How did the Mars satellites "take cover" from the recent comet flyby? After comet Sliding Spring's Oct. 19, 2014 close flyby of Mars, NASA reports that 


  All three NASA orbiters around Mars confirmed their healthy status Sunday after each took shelter behind Mars during a period of risk from dust [from the comet's tail].


"Took shelter behind Mars"?  These spacecraft are in orbit around Mars, constantly in motion, so they can't very well just pack up and move to the other side of Mars to avoid getting hit with the dust.

So my questions are: What exactly qualifies as taking shelter behind Mars?  Does it simply mean being on the other side of the planet from the tail of the comet as dust from the tail passed Mars?

How did NASA get the (constantly in motion) spacecraft to hide behind Mars?  I assume that the orbits had to be modified in advance of the comet's arrival. I picture some orbital modifications that changed the phase of the orbits of the satellites so that they would be on the other side of Mars as the dust flew by, perhaps also making the orbits more elliptical with the apsis behind Mars, allowing the satellites to spend additional time in Mars' protection.

Will these orbital modifications affect the ability of the spacecraft to fulfil their primary missions as they relate to Mars?  I am not asking whether dust from the comet will affect the quality of the instruments on the spacecraft (indeed, this whole question focuses on steps NASA is taking to prevent that from happening) but whether the altered orbits the spacecraft have after their orbital modifications will affect their ability to pursue their primary missions (such as flying over different parts of Mars than was originally intended, which may or may not be beneficial to answering the questions the spacecraft were originally launched to answer.  Although scientists are nothing if not resourceful and I'm sure any data that come from the spacecraft, whether intended or not, will lead to important discoveries).
I've written an answer or two dealing with this within the last week, and I can't believe I never thought about this. Great question.


  These spacecraft are in orbit around Mars, constantly in motion, so they can't very well just pack up and move to the other side of Mars to avoid getting hit with the dust.


Believe it or not, they can just pack up and move. All they have to do is alter their orbital speed - i.e. either slow down, speed up, or change their orbit so they will automatically go a different speed, and so be in a different place. They execute a series of orbital maneuvers to ensure that they are in the part of their orbit that temporarily carries them behind Mars. Orbital maneuvers aren't hard; practically any spacecraft can modify its orbit slightly, if need be. According to this,


  The orbiters will be repositioned to a “safe zone,” where Mars will actually help shield the satellites from the cloud of dust particles.


In other words, the spacecraft will be moved so they are on the opposite side of Mars as the comet. They can't be there for long - after all, they orbit Mars extremely quickly. But, fortunately, they don't have to be there for long. As this says,


  The satellites will remain in this region for a period of up to 40 minutes, until the threat has subsided.


Why do short a period? Well, the comet will only be in the area for a short time: NASA estimates the worst part will last roughly 20 minutes. After that, the danger will have subsided. Any particles that could damage the spacecraft will be long gone.


  Did these orbital modifications affect the quality of research these satellites will be able to perform in relation to Mars?


I can't find a source for the answer to this, but I would expect there wouldn't be any problems. You should be able to undo any maneuvers that would change the orbit of the craft.