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When New Horizons arrives at Pluto next year, how much time will we get to image and explore (measurements, spectra-analysis) the Pluto system? Any ideas if we will see (or be able to count) Pluto's moons?

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Observations will be of two different kinds. Distant, effectively telescopic observations starting months in advance of the flyby and then a torrent of activity during the actual flyby. There will definitely be a chance to observe the known moons of Pluto, which is part of the plan. You can see a timeline of Pluto encounter events on wikipedia. There will be a period of over two months where New Horizons will still be approaching the Pluto/Charon system but the resolution of the images returned will be better than what Hubble is able to achieve (which, admittedly isn't great), and will thus be able to make valuable observations over that period. The actual flyby will occur over a fairly short period of time. The probe will be traveling so fast that it travels over a million kilometers per day. By the time we receive telemetry from the probe indicating it is 100,000 km away from Pluto (nearly 10x the minimum flyby distance) the probe will have already completed its flyby and be 100,000 km on the other side of Pluto. So the vast majority of the most valuable scientific observations will be entirely pre-programmed and we won't see the data until many hours or days later.

As far as the smaller moons, New Horizons should be able to image them to a sub-km resolution.

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Wikipedia has a time table of the mission.

July 14, 2015 it will pass within 13695 kilometers traveling at abour 14 km/s. Based on that info I did this drawing:

enter image description here

The Wikipedia article also has a nice description of what they'll be doing during the Pluto Fly By

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