Praise be to Allah.
Firstly: the correct opinion which must be acted upon is that indicated by the words of the Prophet (peace and blessings of Allaah be upon him): “Fast when you see it [the new moon] and stop fasting when you see it, and if it is cloudy then complete the month with thirty days.” That is the basis for determining the beginning and end of Ramadaan by the actual sighting of the new moon. The Islamic sharee’ah with which Allaah sent our Prophet Muhammad (peace and blessings of Allaah be upon him) is universal and everlasting, until the Day of Resurrection. (It is valid for every time and place, whether worldly sciences are advanced or otherwise, whether there are telescopes etc. or not, whether there are any people in the population who are able to do astronomical calculations or not. Sighting the moon is something that people of all places and times can do, unlike calculations which may or may not be within the reach of the people, and telescopes which may or may not be available)
Secondly: Allaah knew what progress would be made in astronomy and other sciences, but despite that He still said (interpretation of the meaning): “… So whoever among you sights (the crescent on the first night of) the month (of Ramadaan), he must observe sawm (fast) that month…” [al-Baqarah 2:185] and the Messenger of Allaah (peace and blessings of Allaah be upon him) said: “Fast when you see it [the new moon] and stop fasting when you see it…” So the beginning and ending of Ramadaan are connected to the actual sighting of the moon, and not to the calculations of astronomers, even though Allaah knew that the astronomers would develop their science to the point where they would be able to calculate the positions of the stars and heavenly bodies. The Muslims are obliged to follow what Allaah has prescribed for them through His Prophet (peace and blessings of Allaah be upon him), basing the beginning and end of the fast on the sighting of the moon. This is like the consensus of the scholars, whoever differs with this and relies on calculations is shaadhdh (“odd”) and his opinion cannot be relied on. And Allaah knows best.