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Q:

When a rocket leaves the atmosphere and enters a zero gravity environment, the gravity of the sun must affect the rocket or shuttle at some point, and wouldnt the sun begin to pull the shuttle towards it?

A:

When a rocket leaves the atmosphere it is still affected by gravitational pull, so it is not exactly in a "zero gravity environment". If there were no gravity in space, the Earth would not orbit the sun. When we are talking about gravity in space, we refer to it as a "micro-gravity environment". Gravity is still acting on everything, but not as much. You are correct that when the rocket moves further from the Earth and closer to the sun, it will be more affected by the sun's gravitational pull. 

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