# Why do the geosynchronous TDRS satellites have this distribution of inclinations?

From the Tracking and Data Relay Satellite System (TDRSS) option on Celestrak's NORAD Two-Line Element Sets Current Data TLE page I've compiled current TLE inclinations for "on line" TDRS spacecraft:

COSPAR       NORAD    Launched       Spacecraft   current inc(degs)
1988-091B    19548    1988-09-29     TDRS 3          14.4405
1991-054B    21639    1991-08-02     TDRS 5          14.5306
1993-003B    22314    1993-01-13     TDRS 6          14.0861
1995-035B    23613    1995-07-13     TDRS 7          15.0545
2000-034A    26388    2000-06-30     TDRS 8           7.9573
2002-055A    27389    2002-03-08     TDRS 9           5.8274
2002-055A    27566    2002-12-05     TDRS 10          5.5187
2013-004A    39070    2013-01-31     TDRS 11          5.0141
2014-004A    39504    2014-01-24     TDRS 12          5.6613


While not included (yet) on that page, according to this answer TDRS-13 has entered into service, and can still be found in Celestrak:

2017-047A    42915    2017-08-18     TDRS 13          6.7494


While these satellites are in geosynchronous orbits, they would not be considered geostationary orbits; their large inclinations result in analemma-shaped ground tracks, and the antennas of TDRS ground-segment stations would need to trace their daily, nearly North-South movement in order to remain in contact.

This is probably not a big deal, considering the attention and resources already necessary to maintain this critical segment of so many ongoing missions.

But I am curious about the inclinations themselves. There is a cluster (TDRS-3 through TDRS-7) between 14 and 15 degrees, and another (TDRS-9 through TDRS-13) between 5 and 6 degrees. The newest member TDRS-13 is and has been close to 7 degrees for a while now, and TDRS-8 is near 8 degrees.

Question: Are these inclinations specified, station-kept and optimal in some way, or do they simply reflect non-station-kept "inclination creep" that is a natural phenomenon for circular orbits at this distance, or is there another way to uderstand this distribution in TDRS inclinations?

note: I understand that without station keeping, inclination will increase. But I'm looking for more of an answer than "It might be...". Considering that TDRS-13 started at almost 7 degrees suggests that it might be more than just an intrinsic lack of North-South station-keeping ability.

above: Screen shot of TDRS-13 analemma-shaped ground track from N2Y0.

below: Inclination and mean motion (rev/day) for TDRS satellites from the first TLE found in the first week of each calendar year. Dot for TDRS13

• I can't offer more than "it might be" but here goes a comment -- It might be to increase the chances of seeing one of the TDRS birds. If some mission parameter or vehicle obstruction blocks the angle towards the equator, it's OK because some satellites are above and below it. – Saiboogu Mar 28 '18 at 18:49
• @Saiboogu that sounds like a great hypothesis. – uhoh Mar 28 '18 at 22:57
• @Saiboogu take a look at the angles between each pair of TDRS-5, -10, and -11 which are located near each other in longitude, as shown in this answer. It certainly looks (a little bit) like 3-phase AC to me! I think you should pursue this further. – uhoh Mar 29 '18 at 9:49
• I've got some sit-and-wait errands coming up, I'll see if I can do some reading and find some more clues. – Saiboogu Mar 29 '18 at 14:07
• @Saiboogu finished with your errands yet? I've added a bounty. – uhoh Apr 15 '18 at 11:24

Please have a look at the "Onboard Orbit Computations" chapter on the "ACS Without an Attitude" book by Harold L. Hallock Gary Walter David G. Simpson Christopher Rouff.

## Part of section 9.2 - HST Onboard Orbit Models

...In fact, only five parameters are needed to specify the motion of the center of the box in GCI (which we’ll call TDRS from now on for simplicity), namely the TDRS inclination (ideally zero), the TDRS longitude (assigned by the Network Control Center (NCC)), the distance from the Earth center to TDRS (nominally 42,164km), the TDRS rotation rate in the GCI frame (nominally the Earth’s rotation rate), and the Greenwich Hour Angle at the epoch time (i.e., the right ascension of the Greenwich Meridian at the epoch time). The TDRS inclination actually does require calculation because the TDRSS inclinations are allowed to drift up to $14^o$ to conserve fuel and because of the time dependent nature of the GCI reference frame arising from the precession and wobble behavior of the Earth’s spin axis relative to the “fixed stars”. Operationally, exploiting the Flight Dynamics Facility’s (FDF’s) inclined-center- of-box (ICOB) modeling, the advantage of using this so-called “TDRS on-a-stick” model is that the number of uplink parameters (including the epoch time) is reduced from 15 to 6 (the 5 specified above plus the epoch time), and the frequency for parameter updates is reduced from once a week to once a year, or less

Another source of information can be found here on page 18-19.

TDRSs have the same orbit period as a geostationary satellite but are not North/South Station-kept and therefore their orbits may be highly inclined.

• They move in a figure-8 pattern as viewed from earth.

• Inclination evolves between 0 to 15 degrees.

• High inclination provides visibility to South Pole.

• Inclination is not managed as part of stationkeeping.

I think that inclination have a minor effect on the constellation performance. By not controlling inclination the satellites reduce the required amount of propellant dramatically and increase the constellation lifetime.

• Excellent, thanks! I've added a more detailed plot of the inclinations versus time to the question. While conservation of propellant is certainly much of the answer TDRS 8, 9, 10, 11, 12, and 13 were all started somewhere between 6 and 8 degrees of inclination. Curiouser and curiouser! en.wikipedia.org/wiki/Geostationary_orbit#Orbital_stability – uhoh Apr 17 '18 at 13:43
• If your system already accepts drift up to 14° for conservation of propellant, it seems to make sense to accept the same precision on deployment targets as well, especially since initial inclinations begin higher and you're aiming to reduce them as close as necessary to zero. Necessary is <14° evidently. – Saiboogu Apr 17 '18 at 14:32
• @Saiboogu the answer will turn out to be much more interesting than that. – uhoh Apr 17 '18 at 14:43
• Thanks for the additional source. It confirms there is no North-South station-keeping, but doesn't directly address the reason "why" (e.g. nothing about fuel). However it does provide the statement that TDRS can be used by stations at the South Pole, and I'd say that that's more than a "minor effect on the constellation performance". It seems to be a major effect on performance, as standard GEO orbit is invisible from the poles! It's not yet clear if that addresses the "Why?" but it certainly is a positive impact! – uhoh Apr 22 '18 at 1:34