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If two comets come together in space, will they bounce apart, merge into a single body, or could they travel together through space, either touching or orbiting a shared center of gravity?

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Yes, binary comets do exist; 8P/Tuttle is an example of one. This is an area that is still being heavily researched, but it is suspected that binary comets can "form from collision, mutual capture, or fission."

In recent news, the comet 67P/Churyumov-Gerasimenko has also been revealed to be a contact-binary by the European Space Agency's Rosetta spacecraft.

Sources:

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I have a strong feeling that fission would likely be the most likely method of formation due to the absence of any significant gravitational influence of the comets in the solar system and the relative sparseness of the comet/Oort cloud distribution. –  Takku Jul 16 at 20:38
    
@Takku It is possible that some of them formed from collision or mutual capture in the earlier stages of Solar System formation when there was a relatively thicker cloud of ice and dust: solarsystem.nasa.gov/faq/index.cfm?Category=Comets#q3. –  called2voyage Jul 16 at 20:41
    
Oh, yeah of course. I had not considered that they would have 'formed as binaries' and not become binaries after forming. Thanks for pointing it out. –  Takku Jul 16 at 20:45
    
Here is an interpolated video of Churyumov-Gerasimenko. –  fibonatic Jul 18 at 19:04

If the closing velocity is very low, the energy can be dissipated by the comets squishing. You might consider them to merge into one comet in that case. You might be left with a bunch of pieces all travelling together. Whether you call that one comet, the original two, or many is a matter of terminology-all the stuff will be in the same orbit.

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