Orbit Update July 12

So, I figured out why fishy numbers were arising in the calculation. 

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. 

When a orbit is deemed sun synchronous, it's typically in the context of making the nodal precession (or nodal rate regression) equal to 0.98 deg/day, or 360 degrees in a year. With nodal precession, this means that the line of nodes  (connecting the ascending and descending node) precesses at the rate given by the equation located above. 

Sun-synchronous orbits are always defined in the context given above.

However, changing the parameters a,e, and inclination does not just affect the rate of nodal precession. There is a second type of precession that explains what has been happening to these orbits - apsidal precession. Apsidal precession is the rotation of the line of apsides (line connecting apogee and perigee). The rate of apsidal precession, dw, is given by the equation above. 

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. 

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. 

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. 

So, given that the best solution would be to make precession such that  dw +dtheta = 360, I wrote a script to extract those orbits, limit them to those with perigees above 400 km, and extract the one with the slowest angular rate (typically the farthest, most eccentric one). This is the candidate:


Further, I decided to find an orbit that minimized apsidal precession, made dtheta = 360, and existed at 23 degree inclination. This is the candidate:

The question still stands that even if a 23 degree solution doesn't exist due to this problem, there are still several orbits that were analyzed that, when combined or launched at the same time, can provide consistent coverage above 5 minutes for the entire year.