So, I figured out why fishy numbers were arising in the calculation.
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Expectation:
We should expect that when an orbit is put at an inclination of 23 degrees, and the orbit's parameters (a,e,inclination) are set such that the orbit is sun-synchronous, the time spent in the Earth's shadow (for access and observation of the satellite payload) should be the same throughout the year.
In other words, if an orbit is in the plane of the ecliptic, at 23.5 inclination, the satellite should spend some fraction f of its orbit in the Earth’s shadow, independent of time of year.
Problem:
We are not seeing this expectation fulfilled.
Solution:
There is a misunderstanding regarding how the orbit precesses with regard to nodal and apsidal precession.
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There are two interesting notes to point out about apsidal precession.
The first is shown by the text below. This text is taken from " The Apsidal Precession for Low Earth Sun Synchronized Orbits" by Cakaj et. al 2015. It basically confirms that Molniya orbits typically have no precession, due to the inclination they are placed at. While this is not necessarily applicable to our "time spent in Earth's shadow" case (where we want consistent times throughout the year), it is something interesting to note.
The second thing is that sun-synchronous orbits are commonly designed with circular orbits in mind. Therefore, a drift in the line of apsides is less noticeable, since a circular orbit has no eccentricity. It still has apsidal precession, but it's impact is not visible in the satellite's orbit, as you'll see below.
Take this orbit (a= 15100 km, inc =169.27 deg, and ecc = 0.55) above for example. Initially, we did not understand why this orbit did not complete it's full 360 trajectory, even though it satisfied the sun-synchronous (Nodal precession) equation.
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There is one issue, though. When we do these calculations subtracting the the for known sun-synched circular orbits though, my hypothesis isn't consistent.
