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I understand that by sending objects to a specific altitude you can get an object to orbit over a specific area of the Earth. What I have a tough time grasping is how that object then follows the shifting tilt of the planet? Does the tilt shift the entire gravitational field and pull everything with it? Are boosters used?

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Geostationary satellites don't. Or more actually - the tilt remains the same in relation to "distant stars" and the seasons on Earth - the latitude at which Sun is in zenith - changes due to Earth circling the Sun, while its tilt remains fixed.

Meanwhile, the GEO satellites' orbital plane remains just the same in relation to "distant stars" - and as result, to Earth (identical to Earth's equator plane) - and so, it follows the tilt.

It doesn't follow Earth's axial precession which is the actual change of its tilt, but with one precession period of 26,000 years, the necessary corrections using RCS are minimal.

Now, for geosynchronous satellites - these remain at about the same altitude, but aren't necessarily in the same plane as Earth's equator. In their case the plane remains fixed as well, but being shifted so, the satellites travel north-south and back, with one travel cycle per day.

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