r₂

The "magic" part of Gauss' Method is that it can estimate the orbit radius of a satellite when only three angular observations have been made (no distance measurements at all). This step will perform this "magic"; unfortunately, we will need to find all of the real positive roots of an 8th order equation, which takes the form:

Ax⁸ + Bx⁷ + Cx⁶ + Dx⁵ + Ex⁴ + Fx³ + Gx² + Hx + K = 0

It is a little easier in this case, since the coefficients B, D, E, G and H are all 0 (from the previous step). Therefore, the 8th order equation for Gauss' Method is:

A(r₂)⁸ + C(r₂)⁶ + F(r₂)³ + K = 0

where r₂ is the estimated geocentric range of the satellite corresponding to the second observation.

We can use some common sense and intuition to determine what the r₂ distance could be. Most satellites orbit at geocentric distances of between 6,700 and 45,000 km. This means that one (or more) of the roots of this equation fall between these two values. Sometimes you can have multiple roots falling between these two values which means that several orbits are possible (but not always physically possible, so be careful). This is the main difference between Orbit Propagation and Orbit Determination. In Orbit Propagation, there is an exact answer based on mathematics. In Orbit Determination, there is only an estimated answer because the measurement errors will never let you have a perfect orbit solution that will exactly fit your tracking data (no matter how much of it you have).

The simplest method of finding r₂ is to use a spreadsheet and "guess" values until the equation does equal 0 (or nearly so). This is a very manual approach and is not recommended if you wish to determine many orbits. For this example, the manual approach was used and it found only one root in the range from 6,700 km to 45,000 km:

r₂ = 7,194.4100 km

If we could interpret this as an average geocentric distance, the semi-major axis of this orbit could be construed as being about 816 km; which is appropriate for a LEO satellite. However, if the eccentricity of the orbit is high, this value might be near a perigee of a much larger orbit (such as a Molniya-type). The only way to know for sure is to press onward and determine the orbit elements themselves.

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