Certain quests, such as Petra's The Taken War, charge you with killing Taken Champions. Here's everything you need to know about the Earth encounters.
Below you'll find specific details on Earth's Taken Champion and tips on defeating them. If you're looking for an explanation of how the Taken Champions spawn and the mechanics of the instances, go here.
Irxori, Lost to Oryx
Irxori appears in Skywatch. Head down to the Cosmodrome, then towards the building topped with the dish. Go inside, go through the Lunar Complex and you'll appear in the Skywatch area.
This is one of the easier Taken Champions to defeat as the area in which the Lieutenants spawn is relatively enclosed. You'll probably still need a three-person fireteam, though. Just shoot him in the head, but don't get too close. He's got a large flaming gun with your name on it.
Darnu, Horn of Oryx
Darnu appears in The Divide. Head right from the patrol spawn area and go up the red stairs. Head through Dock 13 and you'll emerge in the right place.
You shouldn't have too much trouble with Darnu. The Divide is usually well-populated with other Guardians and there are plenty of vantage points with cover surrounding the central area. Just watch out for the snipers, keep out of the way of the Captains' black bubbles and thump the boss dude in the head. He won't last long.
Stirok, Banner of Oryx
Stirok appears in Rocketyard. Carry on from The Divide area through a roadway to the right. You'll come out in the correct place.
The Stirok encounter is the hardest of the three Earth Champions by approximately 43,000 miles. This guy doesn't go down easily. You must approach this as a three-person fireteam. Trust us on that.
Your first problem is that the area in which the Taken spawn is large and divided into three distinct sections, each obscured from the rest. The first is the section before the wall in which you first enter the Rocketyard. The second is the area past the first tunnel, and the third is the far side, again past a wall. You'll need a Guardian in each to ensure you kill the Lieutenants in time to stop the encounter ending.
But the toughness truly begins when Stirok spawns. Your chances of success here really depend on the Light level of the team, how many other people get involved in the fight and where the boss appears. Stirok is hard. You will have to shoot him in the head many times to kill him.
A good place to stand is on the raised passage between the first and second areas. It's on the left, straight ahead of you as you first enter the Rocketyard. If he spawns in the far, third area, try standing on the edge of the second area on the left and shooting across.
You need somewhere where you can hang back and land headshots while remaining relatively undisturbed. If you have two people standing up here, one can deal with any Captains that jump up onto the platform while the other shoots Stirok without pausing. Be careful of the fan of flames he throws out: it's lethal.
Note: you can't stop shooting the boss for more than a few seconds. You have to stay away from the adds and just pummel him in the face. It's the only way. Award yourself with a nice cup of tea when he finally goes down.