At Aranarth's urging, Círdan sent a ship into the north to recover his father, but it never returned. It was later learned that the ship had rescued Arvedui, but had been lost with him aboard in the icy northern seas. So Aranarth inherited the leadership of the scattered and diminished northern Dúnedain, but he took the title Chieftain, rather than King, since his father's realm was lost.
The Dúnedain were to have their revenge soon after, though, with the arrival of the Gondorian prince and general Eärnur, the son and heir of Eärnil II. He sailed into Lindon, and then marched eastwards to rout the armies of Angmar. Of Aranarth's activities during this campaign we know nothing, but all available evidence suggests that he was in Lindon when Eärnur arrived there, and so it seems very likely that he marched with Eärnur's forces and saw the defeat of his ancient enemy with his own eyes.
Two years after his father's death, Aranarth took the title of Chieftain of the Dúnedain, a title he held for 130 years. He was succeeded by his eldest son, Arahael.
The date of Aranarth's birth appears only in The History of Middle-earth volume 12, The Peoples of Middle-earth. It cannot therefore be considered completely reliable.
At the time of his birth, Aranarth was the heir apparent to the kingdom of Arthedain, so while he never actually inherited the kingdom, his royal name makes some sense. His father Arvedui had been prophesied to be the last of Arthedain'sKings, so perhaps he hoped to counteract this prophecy by giving his son Aranarth a name that emphatically referred to him as a future King of Arthedain. If so, Arvedui's hope was cheated: Arthedain was lost and Aranarth became the first of the line of Isildur not to hold the title 'King'.