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I want a certain asteroid of my system to break into pieces as it approaches a planet, having one fragment impact the planet itself, another fragment slingshot and impact the planet's moon and the rest either burn up or accelerate enough never to be seen again.
Would gravitational forces take care of this on their own if the asteroid's trajectory passed close enough to the planet? Would the asteroid perhaps need to dip a bit into the atmosphere in order to break?
Yes, what you are asking is exactly the definition of the Roche limit
In celestial mechanics, the Roche limit, also called Roche radius, is the distance from a celestial body within which a second celestial body, held together only by its own force of gravity, will disintegrate due to the first body's tidal forces exceeding the second body's gravitational self-attraction. Inside the Roche limit, orbiting material disperses and forms rings, whereas outside the limit material tends to coalesce. The Roche radius depends on the radius of the first body and on the ratio of the bodies' densities.
The Roche limit typically applies to a satellite's disintegrating due to tidal forces induced by its primary, the body around which it orbits. Parts of the satellite that are closer to the primary are attracted more strongly by gravity from the primary than parts that are farther away; this disparity effectively pulls the near and far parts of the satellite apart from each other, and if the disparity (combined with any centrifugal effects due to the object's spin) is larger than the force of gravity holding the satellite together, it can pull the satellite apart.
Some real satellites, both natural and artificial, can orbit within their Roche limits because they are held together by forces other than gravitation. Objects resting on the surface of such a satellite would be lifted away by tidal forces. A weaker satellite, such as a comet, could be broken up when it passes within its Roche limit.
Atmospheric drag can of course cause part of the fragments to fall on the planet.
You can refer to this question on Astronomy.SE for more info and examples.
Yes. Real-world example: https://en.wikipedia.org/wiki/Comet_Shoemaker%E2%80%93Levy_9
Of course this depends on the cohesive strength of the particular asteroid, but a lot of them seem to be loosely-bound "rubble piles": https://en.wikipedia.org/wiki/Rubble_pile
