There is nothing wrong with using modern equipment to sight the moon
Question

Is it permissible to rely on the calculations of the astronomical observatory regarding the beginning and end of a month? Is it permissible for the Muslim to use the modern equipment to sight the moon? Or must it be seen by the naked eye?.
Praise be to Allah.The shar’i method of proving that the new month has begun is for the people to sight the crescent moon. That should be done by people whose religious commitment and strong eyesight are to be trusted. If they sight it, then it is obligatory to act upon this sighting, by fasting if it is the new moon for the month of Ramadaan and by breaking the fast if it is the new moon for the month of Shawwaal.  
It is not permissible to rely on astronomical calculations if the moon is not sighted, but if there is a sighting, even if it is done by an astronomical observatory, then it counts, because of the general meaning of the words of the Prophet (blessings and peace of Allah be upon him): “When you see it then fast, and when you see it, then break the fast.” 
With regard to calculations, it is not permissible to act upon them or to rely on them. 
 
With regard to using a telescope to see the crescent, there is nothing wrong with it, but it is not necessary, because the apparent meaning of the Sunnah is that we should rely upon ordinary sighting and not anything else. But if it is used and the moon is sighted by a trustworthy person, then this sighting may be acted upon. In the past, the people used that because they used to climb up the minarets on the night before the 30
th of Sha’baan and the night before the 30th of Ramadan and look for it by means of a telescope. 
Whatever the case, once it has been proven that it has been sighted by any means, then it becomes obligatory to act upon this sighting, because of the general meaning of the words of the Prophet (blessings and peace of Allah be upon him): “When you see it then fast, and when you see it then break the fast.” End quote. 
Shaykh Muhammad ibn ‘Uthaymeen (May Allah have mercy on him). 
Fatawa ‘Ulama’ al-Balad al-Haram (p. 192, 193) 
we have quoted a fatwa from the Standing Committee for Academic Research and Issuing Fatwas concerning this issue in the answer to question number 1245, in which it says: It is permissible to use telescopes etc. to sight the moon, but it is not permissible to rely on astronomical data to confirm the beginning or end of the blessed month of Ramadaan. End quote. 
See: Fatawa al-Lajnah al-Daa’imah (9/99) 
Thus it is clear that those who claim that our scholars say that it is haram to use modern equipment to sight the moon and say that it is obligatory to sight it with the naked eye are lying and making things up. 
We ask Allah, may He be exalted, to show us the truth as truth and enable us to follow it, and to show us falsehood as falsehood and enable us to avoid it. 
And Allah knows best.