GPS satellites all orbit at the same altitude. According to the FAA,
GPS satellites fly in circular orbits at an altitude of 10,900 nautical miles (20,200 km) and with a period of 12 hours.
Why were these orbits chosen?
You can determine your position using satellites that are not in circular orbits, but with circular orbits all the satellites are on the surface of a sphere, and this insures that certain difficulties don’t occur [1].
To maintain a circular orbit, the velocity is determined by the altitude, and this in turn determines the period. The period T is given by
where μ is the “standard gravitational parameter” for Earth, which equals the mass of the earth times the gravitational constant G.
The weakest link in calculating of T is r. The FAA site says the altitude is 20,200 km, but has that been rounded? Also, we need the distance to the center of the earth, not the altitude above the surface, so we need to add the radius of the earth. But the radius of the earth varies. Using the average radius of the earth I get T = 43,105 seconds.
Note that 12 hours is 43,200 seconds, so the period I calculated is 95 seconds short of 12 hours. Some of the difference is due to calculation inaccuracy, but most of it is real: the orbital period of GPS satellites is less than 12 hours. According to this source, the orbital period is almost precisely 11 hours 58 minutes.
The significance of 11 hours and 58 minutes is that it is half a sidereal day, not half a solar day. I wrote about the difference between a sidereal day and a solar day here. That means each GPS satellite returns to almost the same position twice a day, as seen from the perspective of an observer on the earth. GPS satellites are in a 2:1 resonance with the earth’s rotation.
(But doesn’t the earth rotate on its axis every 24 hours? No, every 23 hours 56 minutes. Those missing four minutes come from the fact that the earth has to rotate a bit more than one rotation on its axis to return to the same position relative to the sun. The four minutes a day add up to a quarter of a year, which is why we have leap years. More on that here.)
[1] Mireille Boutin, Gregor Kemperc. Global positioning: The uniqueness question and a new solution method. Advances in Applied Mathematics 160 (2024)