It is understandable that the length of a solar year can be found out using the time of solstices. But how did they find the length of a year with respect to the stars?

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    $\begingroup$ Hi, welcome to hsm. According to Wikipedia, "A sidereal year is the time taken by the Earth to orbit the Sun once with respect to the fixed stars... Before the discovery of the precession of the equinoxes by Hipparchus in the Hellenistic period, the difference between sidereal and tropical year was unknown. For naked-eye observation, the shift of the constellations relative to the equinoxes only becomes apparent over centuries or "ages"". $\endgroup$ – Conifold Jan 9 at 21:37

By looking at stars position with respect to the point of intersection of the ecliptic and equator (this is a position of the Sun at the equinox). Or which is the same at the position of this point among the stars. This position varies slowly (this is called precession of the equinox), and it was discovered by Hipparchus. Hipparchus work did not survive and it is known from Ptolemy. Since the precession is rather slow, about 1.4 degree per century, to detect it one has to compare two observations at the great time difference (several centuries). Since the time of Hipparchus this was done by many astronomers arriving to more and more precise values of the sidereal year.

Of course the practical problem is that Sun and stars are not visible simultaneously. But fortunately for astronomers: a) we have Moon, and b) Sun and Moon move approximatelty on the same circle. This permits to measure the angular distance between the Sun and Moon when they are both visible and then from the Moon to a star when they are both visible. For this reason, Ptolemy has to consider the motion of the Sun first, then the motion of the Moon and only after that he explains how stars positions with respect to the equinox point are measured. There is a continuing discussion lasting more than 100 years on the question "who made the first star catalog, Hipparchus or Ptolemy?" But Ptolemy's work is the only surviving ancient work where all this is explained in detail.

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  • $\begingroup$ Priority is often a matter of dispute and there is a fringe opinion that Aristarchos already knew about the difference (Rawlins D., ui.adsabs.harvard.edu/#abs/1999DIO.....9...30R/abstract ). Neugebauer opposed the view that the knowledge came from Babylonians. Anyway 'primitive' people knew that the rising of some constellation slowly becomes a less adequate marker for a season and this what is explained today as precession. $\endgroup$ – sand1 Jan 10 at 11:11
  • $\begingroup$ In such disputes I usually side with Neugebauer. $\endgroup$ – Alexandre Eremenko Jan 10 at 12:28

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