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Since GPS satellites need to face the Earth, they might execute momentum-unloading from time to time using pairs thrusters so as to produce torque but minimal change in the orbit, but I am guessing that the perturbations are so low in MEO and the way the system works mathematically is such that they would never have to make orbital maneuvers once deployed.

Is that correct? Or do they from time-to-time do some kind of station-keeping in order to remain well-spaced and distributed within each of their orbital planes and slots? If they do, wouldn't they have to deactivate the satellite's GPS coverage until a new ephemeris could be determined and uploaded?

(see this comment)

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Yes, GPS satellites do execute station keeping maneuvers. Primary purpose is to keep them within the desired repeating ground track, which leads to a maneuver every year or so for each satellite. I believe primary source of disturbances at that altitude are geodetic variations in gravity field. The maneuvers are needed less than once a year on average. Of course, additional maneuvers may be required if there is a significant constellation issue (e.g. two satellites going dead). [Source: my office neighbor who flew GPS satellites for the USAF]

When a maneuver is about to happen, the USCG and GPS ops center issues a Notice Advisory to Navstar Users (NANU; I wonder if someone was a Mork and Mindy fan) informing users of an impending FCSTDV or forecasted delta V. You can find actual NANUs at the USCG Navigation Center. An example of a recent FCSTDV NANU shows that the predicted outage is usually for 12 hours.

I'm not a GPS expert, but my understanding is that an FCSTDV is a "scheduled outage" and the satellite will not be available during that period.

My friends at AGI keep a nice dashboard of GPS satellite outage information.

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    $\begingroup$ This is excellent! Thanks for putting together this collection of sources and information. I'll dig in over the next few days. $\endgroup$ – uhoh Sep 25 '18 at 14:01

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