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I understand that astronomers use right ascension and declination for the position of stars. However both these values strike me as very anthropocentric and, more importantly, unreliable over time as I understand that the Earth wobbles a bit over time so the celestial equator shifts and I guess Sun's rotation around the galaxy and Milky Way's own trajectory should play a role in depreciating the accuracy of those values over time.

So my questions are:

[1] how steady are right ascension and declination values for distant stars over time?

[2] is there a more stable coordinate system or do we lack an absolutely fixed frame of reference?

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    $\begingroup$ If you use a fixed epoch date, such J2000.0, right ascension and declination values are a lot more stable. The ICRF reference frame that NASA uses is a mostly fixed coordinate system that does not precess or nutate over time. However, it uses our solar system's barycenter as its central point, so it's not 100% fixed. Note that NASA uses ICRF even for data +-15000 years from now. $\endgroup$
    – user21
    Oct 14, 2014 at 0:09

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The values are more steady for objects close to Ecliptic poles, but in genenral, they change by a few arcseconds every year.

Yes, we lack an absolutely fixed frame of reference.

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