The Prophet (peace be upon him) said: We are an illiterate nation; we neither write, nor calculate. The month is like this and this, i.e., sometimes of twenty-nine days and sometimes thirty days. [Narrated by Al-Bukhari, Book on fasting, Chapter on the words of the Prophet, "We do not write nor calculate", no. 1913; and Muslim, Book on fasting, Chapter on fasting in Ramadan should necessarily be commenced with the sight of the new moon…, no. 1080]
The Prophet (peace be upon him) also said: Do not start the fast or break it until you see the new moon. If the new moon is obscured from you, then complete a full thirty days. [Narrated by Al-Bukhari, Book on fasting, Chapter on the words of the Prophet; "When you sight the new moon (of Ramadan) observe fast", nos. 1906 and 1907; and Muslim, Book on fasting, Chapter on fasting in Ramadan should necessarily be commenced with the sight of the new moon…, no. 1080]
Hence, it becomes clear that the reliable factor in determining the time of Sawm and breaking the Sawm and the rest of months is the sighting of the moon or completing the prescribed period. The mere birth of the new moon is not considered in Shari`ah (Islamic law) to verify the beginning and ending of the lunar month with the Ijma`(consensus of scholars) of trustworthy Muslim scholars, unless it is sighted legally. This ruling is pertinent to the timings of worships. Thus, whoever from among the contemporary scholars contradicts this Ijma`, then he will have his opinion rejected as the Sunnah (whatever is reported from the Prophet) of Muhhammad (peace be upon him) and the unanimous agreement of the Salaf (righteous predecessors) should take priority any other argument. So, the calculations of the sun and moon motions are not considered in this respect as we mentioned before and according to the following evidence:
A person must pay attention to the difference to the new juristic issues which aroused controversy and difference among scholars. Such issues were studied by the Council of Senior Scholars in one of its previous seminars and came to a decision by the majority that the most preponderant opinion is that each country has its own sighting and that they have to resort to their scholars in this regard.
Regarding the saying of those who state that one has to depend on the sighting of the new moon in Makkah in particular, this has no evidence or base at all. In addition, it also means that Sawm would not be obligatory on anyone who sights the new moon in other places unless they sight it only in Makkah.
Source: Ruling on Depending on Astronomical Calculations - Fatwas of Ibn Baaz