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ESA publishes quite a few images from the 67P comet, sent by Rosetta, from various distances away to collect as much as possible and as varied as possible.

As I write, Rosetta is 6 Km away from the comet, and can even take clear pictures of its own shadow on the comet.

The comet does spin though, and dust may prevent Rosetta from getting too close or entering some cone formed by the comet's tail (there are related questions).

How close could Rosetta theoretically get to the comet?

The question may not be so easy to answer, as the comet is not spherical, and its spin may not be planar.

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    $\begingroup$ As long as there is no physical contact Rosetta can come within distances too close to measure. You need to be more clear in what you are asking. $\endgroup$ – GdD Mar 4 '15 at 9:36
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    $\begingroup$ Can we assume the meaning is "how small the orbit perigee or the orbit radius can be?" Interesting question since even if there is no atmosphere, the spacecraft can still be slowed down by other elements. $\endgroup$ – mins Mar 4 '15 at 18:36
  • $\begingroup$ Thank you for your comments. Is it clearer to ask an order of magnitude? Now is the range of the Km. I wonder if it can get within the range of a 100m, for instance. The question stems from the unusual shape of the comet (unusual to me). And yes, I think the orbit perigee is an accurate way to put it. I did not have the proper words in mind. $\endgroup$ – Eric Platon Mar 5 '15 at 11:33
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The comet's gravitational force is so small that Rosetta often doesn't really orbit the comet. When Rosetta's distance from the comet gets too large (~100 km), it has to use its thrusters to adjust its trajectory every so often, and its path looks like a slightly curved triangle.
ESA can choose pretty much any orbit they want. In practice, they'll try to make sure Rosetta is far enough away not to be damaged by the gas and dust outflow. They've got 20 years and a pile of money invested, they don't want to lose the spacecraft. The comet is expected to become more active as it's getting closer to the Sun, the recent pass at 6 km was a "last chance" to get this close. It's unlikely they'll get this close again before the end of the mission.
I haven't been able to get a full list of Rosetta "orbits", but pages like these refer to the 6-km pass as Rosetta's closest approach, and indicate the remainder of 2015 will see longer-distance passes.

For the rest of the current mission plan in 2015, in fact, Rosetta will always conduct flybys, and, based on predictions of increasing cometary activity, can no longer be manoeuvred so close to the comet as to be in a gravitationally bound orbit.

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