got mine when i killed oryx, i dont know if was the "road to kings fall" or "old hungry", but was exactly when i delivered a quest to Eris before complete the raid for first time
Here is a quick explanation in case people were wondering. Some people know this but you still see questions, so I thought I would write up something. Note that this is the only way to get a 310 Artifact (or even a 301+) in the game right now.
1) Finish the eight main story missions.
2) Do the Eris CoO quest to get a 300 artifact (I don't know if this is necessary but it's good to have for now). To do this you do one CoO on your own and then go and talk to Eris and pick up the quest. You will see no Eris quest indicator on your map, but when you talk to her she gives it to you.
3) Do all the quests for the vanguard that have a TTK picture in them. This one:
In these quests you have to beat three ultra bosses in three different locations on earth. The same boss spawns in the same location each time as long as you kill all three of his lieutenants. It's Skywatch, Rocket Yard, and the Divide. It helps to bring a fire team. I believe these Ultra's are on a somewhat long timer and if you don't beat them in time they despawn. You have to do some story missions. And you have to beat six lieutenants on mars too.
4) At a certain point you will need to do the shield brothers strike quest in order to continue the "Taken Wars quests". This is sort of poorly explained in game but the Taken Wars quest will not continue, and then you have to do the shield brothers strike quest and then the Taken Wars quests continue.
5) You do not need to do any of the Taken Wars stuff with Petra and Variks. That is completely separate.
6) Eventually you get a quest called "The Road to Kings Fall". This is it. Do the Summoners Circle story mission.
7) Then the last step in the "The Road to Kings Fall" is to beat all the bosses in the raid. Once you do that you can pick up your 310 Artifact from Eris.
? I always run wildcard during this phase, or "floater" if you will. I always use fusion rifle and encourage all the corner guardians to use fusion rifles as well. Tons of damage, tons of ammo, really good at staggering
the ogres
. Solar grenades, Vortex grenades, and whatever those funny defender grenades are called helps a ton as well (basically any grenade that stays active for a long time)