Why did the last six TDRS satellites start at ~7° inclination on a downward trend that appears to be "repelled" by zero & bounce back w/o crossing?

Question

The most recent six TDRS satellites (8 through 13) were inserted into moderate inclination orbits of roughly 7°, and in all cases their inclinations immediately decreased at rates of about +/-1 or +/-0.5 degrees per year.

I'm curious both about the underlying orbital mechanics and the mission design going on here.

Questions:

1. Why was the decision made to put TDRS 8 through 13 at about 7° in inclination?
2. Why do they always drift downward towards zero inclination, and the ones that get there seem to be "repelled" by zero and instead of crossing zero slow down, stop at a small but finite positive inclination then start back up again. $\text{d} i / \text{d} t$ seems parabolic near the minimum but linear otherwise!
3. Why does the inclination of the most recent three TDRS satellites change by half the rate for the first three? (~0.5 deg/year vs. 1 deg/year for the first three)

Related and potentially helpful answers:

• Why do the geosynchronous TDRS satellites have this distribution of inclinations? (discussion below @Eviatar.E's answer inspired this question)
• Why does TDRS 1's inclination evolve so much differently than that of all the others starting in 1995? (@Puffin's excellent answer is also the beginnings of an answer here!)
• What's the method behind this TDRS triplet inclination "madness"?
• How can I "debounce" these TDRS satellite inclinations? (reconstruct the zero-crossings) (no answer, perhaps should be quickly answered and/or closed?

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The six TDRS satellites (8 through 13) are 26388, 27389, 27566, 39070, 39504, 42915

Data is from the following query in Space-Track.org:

https://www.space-track.org/basicspacedata/query/class/gp_history/NORAD_CAT_ID/26388,27389,27566,39070,39504,42915/orderby/TLE_LINE1 DESC/EPOCH/2000-01-01--2022-01-01/format/tle

script for plotting:

import numpy as np
import matplotlib.pyplot as plt
from datetime import datetime, timedelta
fname = 'TDRS TLEs for inclination.txt'
with open(fname, 'r') as infile:
    TLEs = np.array(infile.readlines()).reshape(-1, 2)
print('TLEs.shape: ', TLEs.shape)
satdict_unsorted = dict()
for L1, L2 in TLEs:
    satid = int(L1[2:7])
    year = 2000 + int(L1[18:20])
    decimal_days = float(L1[20:32])
    # https://stackoverflow.com/a/34910712/3904031
    datetim = datetime(year, 1, 1) + timedelta(decimal_days - 1)
    seconds_since_epoch = (datetim - datetime(1970, 1, 1)).total_seconds()
    decimal_year = 1970. + seconds_since_epoch / (365.2564 * 24 * 3600)
    inc = float(L2[8:16])
    raan = float(L2[17:25])
    ecc = float('.' + L2[26:33])
    n = float(L2[52:63])
    T = 24 * 3600 / n
    info = [year, decimal_days, seconds_since_epoch, decimal_year,
            inc, raan, ecc, T, n]
    try:
        satdict_unsorted[satid].append(info)
    except:
        satdict_unsorted[satid] = [info]
satdict = dict()
for key, thing in satdict_unsorted.items():
    thing.sort(key=lambda x: x[2])
    print('len(thing): ', len(thing))
    print('len(thing[1]): ', len(thing[1]))
    satdict[key] = np.array(list(zip(*thing)))
if True:
    fig, (ax1, ax2, ax3, ax4) = plt.subplots(4, 1)
    names = 'inc (deg)', 'RAAN (deg)', 'Ecc', 'T (sec)'
    T_siderial = 23 * 3600 + 56 * 60 + 4.09
    for key, thing in satdict.items():
        years, inc, raan, ecc, T, n = thing[3:9]
        ax1.plot(years, inc, label=str(key))
        ax1.set_ylabel('inc (deg)')
        ax2.plot(years, raan, label=str(key))
        ax2.set_ylabel('RAAN (deg)')
        ax3.plot(years, ecc, label=str(key))
        ax3.set_ylabel('Ecc')
        # implement a crude rolling median removing TLEs too close together
        d_t = years[1:] - years[:-1]
        d_inc = inc[1:] - inc[:-1]
        keep = d_t > 100. / 3600 / 365.2564 # years
        dinc_dt = d_inc[keep] / d_t[keep]
        yearz = years[:-1][keep] # fudge
        N = 20
        rolling = [np.median(dinc_dt[i:i+N]) for i in range(len(dinc_dt)-N)]
        ax4.plot(yearz[10:-10], rolling)
        ax4.set_ylabel('d inc/dt (deg/year)')
    ax1.legend()
    ax2.legend()
    ax1.set_ylim(0, 10)
    ax3.set_yscale('log')
    ax4.set_ylim(-1.2, 1.2)
    ax4.plot(ax4.get_xlim(), [0, 0], '-k')
    fig.suptitle('TDRS')
    plt.show()

Answer 1

These satellites are on the same 53-year long cycle of RAAN and inclination changes like any geosynchronous satellites whose inclination is not actively maintained. Essentially, it's the combined influence of the Moon, the Sun and Earth's oblateness (J2) that pulls on the orbit.

There's a nice paper about this: Anderson, Paul; et al. (2015). Operational Considerations of GEO Debris Synchronization Dynamics (PDF). 66th International Astronautical Congress. Jerusalem, Israel.

Please check figure 2 on page 4 for a graphical representation of the RAAN/inclination changes of the orbits. I won't add it here due to the unclear copyright.

Why do they always drift downward towards zero inclination, and the ones that get there seem to be "repelled" by zero and instead of crossing zero slow down, stop at a small but finite positive inclination then start back up again.

This is what actually happens - the change in inclination stops, and the change in RAAN speeds up.

di/dt seems parabolic near the minimum but linear otherwise!

That's just a coincidence - there are parts of the cycle that are almost linear, and strongly curved parts at other positions.

Why does the inclination of the most recent three TDRS satellites change by half the rate for the first three?

They are on an orbit with a different RAAN - The older ones were inserted at about -60°, while the newer ones are at only -30°. This puts them closer to the equilibrium point and therefore makes them evolve slower.

Why was the decision made to put TDRS 8 through 13 at about 7° in inclination?

It's always cheaper to launch into an inclined orbit than to a precisely stationary orbit (unless your space port is at the equator, that is). The chosen inclination and slightly negative RAAN assures that the satellite will always orbit at an inclination lower than the initial one for the next 20 years. Only after this time the inclination will be larger with a peak at about 14° - but at this time it's likely decommissioned already.