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NASA has approved the New Horizons mission to 2014MU69 on 1 January 2019 that will require operations at least into 2021 to download the data. However, after that, things get a bit fuzzy - New Horizons will still have a substantial amount of fuel left, so it could theoretically visit another KBO - if one was found close to its trajectory. Even if another object in the Kuiper Belt doesn't turn up to visit, NH is on a trajectory that will take it out of the Solar system.

The Voyager probes are currently (4 September 2017) traveling 17kps and 15.4kps away from the Sun, in interstellar space (beyond the heliopause), and transmitting scientific data. Although it was the fastest vehicle ever launched, New Horizons is "only" going 14.2kps, so it will take quite a bit longer to reach the heliopause than the Voyagers did - but it will get there, and go beyond.

Assuming it doesn't have a fatal crash within the Kuiper Belt, how long could New Horizons continue to return scientific data? Is it limited by its data rate at distance, or by the power available in its RTGs?

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RTG power will drop below the level needed to run the transmitter sometime in the 2030s.

We can still communicate with the Voyagers, which are 3 times further away than New Horizons. NH is more flexible than the Voyagers in what data rates it can use and how much data it can store for later transmission, so it's unlikely that will be the bottleneck.

The bigger question is, what science can New Horizons do outside of planetary encounters? It carries some particles experiments (SWAP, PEPPSI and SDC) that could be useful. SWAP has been used en route to Pluto to monitor the solar wind, this would be useful to keep running. PEPPSI was designed to measure Pluto's atmosphere, I don't know if it can be used in interplanetary space. Same for SDC.

Of course, all of this is guesswork. The New Horizons extended mission is funded until 2021. Operations after that will need another mission extension, for which the NH team will need to compete with other interplanetary missions. The NH team has indicated it's looking for possible targets for flybys after 2014MU69.

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