Is it permissible to rely on astronomical calculations to determine the beginning of the month of Ramadan, or is it necessary to sight the crescent moon with the naked eye?

It is permissible to use observation instruments to sight the crescent moon, but astronomical calculations are not to be relied upon for determining the beginning of Ramadan or Eid al-Fitr, because this has not been mentioned in the Sharia. Rather, the Sharia relies on the actual sighting of the crescent moon for the start of fasting, breaking the fast, and holidays, based on Allah's saying: "So whoever sights [the new moon of] the month, let him fast it," and "They ask you, [O Muhammad], about the new moons. Say, "They are measurements of time for the people and for Hajj." And based on the Prophet's (peace be upon him) saying: "Fast when you see it [the crescent moon] and break your fast when you see it. If it is obscured from you, then complete the number [of days] as thirty." So, whoever does not see the crescent moon should complete the month of Sha'ban as thirty days, unless its sighting is legally confirmed in another country, in which case they must fast based on the opinion of the majority of scholars.