# Differences between Semi Major Axes and ECF Magnitudes for Circular Orbits with J2 Perturbation

I have simulated (in STK) a satellite at near circular orbit, near-equator, with Brouwer Lyddane Mean Long SMA of 6935.31km, inclination 10 degrees, and eccentricity of 0.0001875 (these are solved-for repeat ground track parameters). I have included J2 perturbations only. No third body, no drag, solar decay etc. Just J2.

I noticed that the ECF magnitudes and the semi major axes are not exactly the same, in fact there seems to be some +10km bias offset between the two.

My semi major axes are shown below, with the black line being the osculating one, and the little amplitude teal lines being the mean SMAs (Brouwer Lyddane Mean Long and Short). It is about 6935.31 as I had input it.

Below is the plot of the altitudes and magnitudes, with the LHS y-axis belonging to the ECF position magnitude (geocentric), and the RHS y-axis belonging to the ECF altitudes.

The ECF magnitudes make sense somewhat to me. For example, if I take the ECF magnitude 6926.15km – 6378.14km (Earth’s equatorial radius), I get about 548km, which is what the purple and green lines corresponding to the detic and perigee altitudes show.

However, if my semi-major axes is 6935.31km, I am not sure how that relates to the ECF magnitude, considering that it is almost 10km larger than the mean ECF magnitude even though the eccentricity is almost zero, and so it should be almost circular.

Main Question: What is the relationship between the mean SMAs versus the ECF magnitudes, and why are they so different (10km off; beyond expections!) even for an almost-circular orbit with altitude variations of +/- 600m only?

• A man can drive himself crazy looking at STK output plots... Mar 17, 2020 at 4:21
• different but related and potentially helpful: Brouwer-Lyddane mean semi major axis bias and What's a Brouwer-Lyddane mean semi major axis, or any other, for an orbit in a lumpy gravity field? and also see this answer
– uhoh
Mar 17, 2020 at 6:14
• Thanks for pointing it out! I enjoyed reading your answer and analysis @uhoh, and thank you also for sharing the Python code :) Mar 17, 2020 at 11:47
• Thanks! In my defense, didn't mean to link to any user in particular, just searched the site for either "Brouwer" or "Lyddane" (don't remember which) and linked to related posts.
– uhoh
Mar 18, 2020 at 0:52