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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Apsidal precession and nodal precession can either work with each other or against each other, in that they sum to the total amount of precession an object experiences in one year.
Take .
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, it is something interesting to note. The second thing is that sun-synchronous orbits are commonly designed with circular orbits in mind
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.
When running the numbers on the induced apsidal precession, and comparing it to the nodal precession initially expected, we see that this hypothesis of "the apsidal(dw) and nodal(dtheta) precession affect each other" come into play. The calculation below suggests a total precession of 340 degrees in one year, roughly similar to what was observed above, as the orbit does not quite complete itself.
This was also true of the a = 12717 km, inc =156.5 deg, and ecc = 0.3 observed. The orbit appears to drift 270 degrees by the end of the year, similar to what is calculated for the difference in the apsidal and nodal rate of precession (278 deg. recorded).
Finally, this seemed to be confirmed by STK Support staff, who echoed that I should consider apsidal rate. Everything lines up such that solving the apsidal rate problem is the issue.
There is one issue, though. When we do these calculations subtracting the for known sun-synched circular orbits though, my hypothesis isn't consistent.
