In 340 BC the Byzantines and their allies the Athenians were under siege by the troops of Philip II of Macedon, the father of Alexander the Great. On a particularly dark and wet night Philip attempted a surprise attack but was thwarted by the appearance of a bright light in the sky in the middle of the crescent. To commemorate the event the Byzantines took as their emblem the sky and the crescent.
When St. Constantine the Great moved the seat of the Empire from Rome to Byzantium, he changed the name of the city to Constantinople and took out the star and put the cross instead as you can see in many places on Mount Athos.
The double-headed eagle was only the flag in the late period of the empire but it was never the emblem of it. Only after the fall, the double-headed eagle entered slowly in the conscience of the people as an emblem.
The Ottoman Empire, considering itself the follower of the Byzantine Empire removed the cross and placed the star back and till today this is the flag of the Turkish state.
This photo shows the actual astronomical phenomenon occurring on the Mount Athos over Vatopedi’s main church (katholikon).