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I am pretty confused about the concept of a co-rotating frame? I am specifically referring to the Geocentric Celestial Ref. System (GCRS). Can anyone enlighten me with an example or a counter example as to why the GCRS is or is not? I have read several documents, some say the GCRS is inertial, some say it is non-inertial. So I am pretty stumped.

Any help is great appreciated!

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I do not know whether you figured this out, but I will answer anyway. The Geocentric Celestial Reference System I believe is a Newtonian theory object, not so much a general relativity one. The Geocentric Celestial Reference System can be also found under the name Earth Centered Inertial coordinate system. Technically, it is not quite inertial but can be a very good approximation of an inertial coordinate frame that is used for the description of the motion of satellites orbiting Earth.

There are two parts one can focus on: (1) the origin of the Geocentric Celestial Reference System and (2) its axes.

(1) The Geocentric Celestial Reference System has its origin at the gravitational center of the Earth, so it moves with it, as Earth revolves around the sun. This motion is not uniform along a straight line, but follows an orbit which is roughly an ellipse. That makes the Geocentric Celestial Reference System non-inertial. However, for practical purposes, the direction of Earth's velocity vectors changes very slowly, so for short periods of time Earth moves almost uniformly along a straight line, so you can assume that the system is inertial.

(2) The more important part is that the axes of the Geocentric Celestial Reference System are actually fixed with respect to the surrounding space, i.e. they are fixed with respect to the stars. So they actually do not change direction and orientation as Earth revolves around the sun. So these axes are almost truly inertial (as inertial as we can measure distant objects in the very far periphery of the universe).

So, basically, the origin of the Geocentric Celestial Reference System is not inertial, but the axes are inertial (so no rotation, no fictitious rotational forces). And for that reason it is very easy to correct the equations of motion if in this system whenever necessary and make things precise (you just have to account for the linear acceleration of Earth's center as it revolves around the sun).

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