Praise be to Allah.
What counts with regard to confirming the beginning of the month of Ramadan is sighting of the new moon or the completion of 30 days of Sha’baan if the moon is not sighted. This is what is indicated by the saheeh Sunnah, and the scholars are unanimously agreed on it. Al-Bukhaari (1909) and Muslim (1081) narrated that Abu Hurayrah (may Allah be pleased with him) said: The Prophet (blessings and peace of Allah be upon him) said: “Fast when you see it and break the fast when you see it, and if it is cloudy then reckon the month as thirty.”
Astronomical calculations do not count. The basic principle with regard to sighting is that it should be done with the naked eye, but if the crescent moon is sighted with modern equipment, then this sighting may be acted upon, as stated in the answer to question number 106489. With regard to how the new moon is sighted with the naked eye when it is not seen with telescopes and equipment, this may be due to differences in the place and time of sighting.
Whatever the case, the ruling depends on sighting of the new moon. So long as it has been seen by one or two trustworthy Muslims, then it is obligatory to act upon this sighting.
Shaykh Saalih ibn Muhammad al-Lahaydaan, the head of the Higher Judiciary Council (may Allah preserve him) said: There is a brother called ‘Abd-Allaah al-Khudayri who is famous for sighting the moon and he is one of those who are known to be involved in sighting the moon at different stages, even at times other than the new moon. Some astronomers went to him and met with him in the region of Hawtah Sadeer. And he told me that they estimated that the moon would appear on that night in a certain place according to the calculations they did on their computers, but he told them that it would not appear from the place that they said, because he had seen it before them the night before, and he knew the stations of the moon and from where it would rise every night. Then when the moon appeared, it rose from the place he mentioned and not they place they mentioned. But he excused them because their conclusion was not based on direct moon sighting; rather it was based on calculations with the computers they had. End quote from an interview with him in al-Riyadh newspaper.
And Allah knows best.