When was Bhagiratha born? Are there any attempts to date his period? Because it is said that the incident of the drought occurred when Kapila Nakshatra was seen in the sky.
I think this might be helpful in mapping how long back Bhagiratha existed.
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According to Ikshvaku_dynasty, King Sagara was ancentor of king Bhagirataha. King sagara had 60,000 sons who became ghosts.
King -> Period
Amsumana was already there in the story where Kapiladeva returned the horse and told Anshuman that the sons of King Sagara could be delivered if the Ganges descended to earth and bathed them in her waters.
Kapila is vedic sage and king Sagara is considered to be vedic king. Now Anshumana sons
23 Dileepa ->
24 Bhagiratha ->
These two can have period of either satya yuga or Treta Yuga.
37 Rama -> Treta yuga
Ganga is mentioned in the Rigveda, the earliest and theoretically the holiest of the Hindu scriptures. Ganga is mentioned in the nadistuti (Rigveda 10.75), which lists the rivers from east to west. In RV 6.45.31, the word Ganga is also mentioned, but it is not clear if the reference is to the river.
So we can see that before Rigveda was composed Ganga was there on earth.
Duration of Satya yuga 1,728,000 years
Duration of lifetime of human in Satya yuga around 100,000 years.
Duration of lifetime of human in Tretā-yuga around 10,000 years
Brahma is at number 1 so at number 24 years can be maximum of 840 *1000 years if we consider a person can have son at the age 35 which is in Gruhasthashram as per vedas.
Time passed after beginning of Kali-Yuga (Battle of Kurukṣetra) around 5,114 years
Year of Kaliyuga started 5,114 years
+ Time of Dwapar Yuga 864,000 years
+ Time of Treta yuga 1,296,000 years
= 2165114 Years
Period of Bhagirataha can be before 2165114 years minimum
Edits are invited to improve this answer