math and really ridiculous gear ratios.
so the motion of the planets as viewed from earth are fairly complex but highly predictable patterns. so predictable that you can program a telescope to follow it simply by telling it how to orient itself over time. the machine doesn’t need to be able to “see” to capture a clear image.
the machinery necessary to do this is fairly expensive due to how precise the motion has to be. the telescope needs to be able to make slow, smooth motions constantly across hours. this usually involves substantial gear trains
Tracking is simply moving the telescope at the same rate that the sky rotates, this can be done as described in other comments by alligning the axis of rotation with the axis of rotation of the Earth or by rotating around 2 different axis.
There is also guiding which is used by higher end setups where you use a seperate telescope and camera mounted to the first and actively correct the tracking by ensuring a target/guide star stays at the same point in the guide scope. Observatories normally also have high powered lasers they can use in the event there are no suitable natural guide stars.
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