There are actually several agencies and even companies, that keep track of satellite & debris orbits.
All together they usually mostly “play nice” with each other, share a lot of information. And so around that mutual willingness to share data between agencies and companies, a kind of de facto system has evolved in which everyone mostly stays out the way of everyone else.
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As for which agencies specifically…
Probably the organization that tracks the most satellites and debris, I am guessing, is probably the US Department of Defense’s United State Space Surveillance Network (SSN).
Control and operations of the SSN has recently been transferred over to the new US military branch: Space Force.
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In addition, NASA does a lot of work and tracking on this, with their Orbital Debris Program Office.
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As well, you’ve got NORAD:
North American Aerospace Defense Command.
This is actually a joint US-Canada military command operation that’s been running since the late 1950’s!
Which means I guess they’re probably the oldest organization that does orbital and space tracking. They too give away and publish a lot of orbital tracking data, by issuing regular report-files.
Interestingly, NORAD has expanded their monitoring to including North American aircraft-flight air space, as well as monitoring for drug-trafficking air craft.
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