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A launcher upper stage may not stay in orbit for long, as it does not need to go as high as the payload and thus travel at an altitude at which although it is thin, the atmosphere is still dense enough to slow down the launcher stage. I don't know how long it typically stays in orbit, but I believe it is a few months for LEO.

But, for GTO, the energy involved is higher: when releasing the payload, the upper stage is fast enough so the payload reaches the desired altitude without significant trajectory correction, and thus the upper stage may stay in orbit longer even if it is slowed down (unwanted aerobrake) when at apoapsis.

For how long does it stay in this orbit? Is there any action taken to decrease this duration?

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The upper-most stage of a launch vehicle is in essentially the same orbit as the payload. The stage burns out, and then the payload is separated from the stage. This is typically at a very low relative velocity on the order of meters per second, several orders-of-magnitude below orbital speed, which is measured in km per second. The separation velocity, and usually a post-separation collision avoidance maneuver performed by the stage, are just enough to prevent re-contact on future orbits.

An obvious exception to this are satellites that significantly change their orbit after deployment. The most obvious example is a satellite that is separated in geosynchronous transfer orbit (GTO) and then maneuvers into geosynchronous earth orbit (GEO).

So upper stages, if not intentionally de-orbited (which some are) can stay in orbit a long time. Stages have low ballistic coefficients because of the empty tanks, which will cause them to re-enter sooner than a more-dense satellite, but they can still stay in orbit for years.

Exactly how long depends a lot on the details: the orbit, the ballistic coefficient of the stage, and the solar cycle (high solar activity "puffs up" the atmosphere, which increases drag). Part of obtaining a launch license (in the US) is running a government provided code to calculate the de-orbit time for the payload and any spent stages. The best practice is to ensure everything is de-orbited (actively or passively) within 25 years.

See related answer to Why do malfunctioning satellites come back to Earth?

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