There are so many astronomical evidences given in Ramayana and Mahabharata. It is said that there are about 500+ in Ramayana and 200+ in Mahabharata.
What calendar they followed
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In Ramayan, greater chances are that they followed lunar calendar.
One of the issues with Ramayan is that why Dashahra and Deepavali are celebrated in autumn when Ram ji left for 14 year exile on his birthday which comes in spring. According to solar calendar this is obviously wrong.
One of the explanations of this prevalent practice, as made by author Saroj Bala in book "Ramayan retold with scientific evidences" in "Box 6.3" (https://books.google.com/books/about/Ramayan_Retold_With_Scientific_Evidences.html?id=vReKDwAAQBAJ) is that they may have followed Lunar calendar. Lunar calendar misses solar calendar by a month in about 32-33 months. In 14 years, we'd have missed about 5 months, which roughly matches the discrepancy. (Btw, author doesn't claim the type of calendar is mentioned explicitly in Ramayana. She says ancients followed the practice of adhik maas to continue with lunar solar calendar like today, but somewhere in between someone ignored the adhik maas and moved celebrations to 5 months earlier).