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Any study that involves any object in space other than Earth or how the Earth interact with other objects.
No. Magnetic compasses work based on the Earth's mantic field, in space there is no magnetic field for the compasses to work with. A different system, possibly similar to Global Positioning System (GPS) might work, call it the Universal Positioning System. On certain rocky planets it could work, but some planets don't have a magnetic field, like Mars. So a traditional magnetic compass wouldn't work in space, or at least it won't get you where you want to go.
No. A compass is a useful tool on Earth because the needle aligns with Earth's magnetic field and we know the shape of that field. There would be no such field in outer space.
It interacts through its gravity field. The main interaction is with the Sun which holds it in a stable nearly circular orbit. Charged particles from the sun are "captured" in the van Allen radiation belt (due to the Earth magnetic field).
Earth has gravity so less fuel is needed to move a rocket in space(assume that you mean escape from earth's gravitational field by"from earth")
Space shuttles are not designed to operate outside of a low-earth orbit, where they are protected from deadly radiation by the earth's magnetic field; a mission to Sirius by a space shuttle is not even possible. For more information on the earth's radiation protective field, see the Wikipedia article:Van Allen radiation belt
It likely depends on the specific program and sub-field you select within biology and/or earth and space science. My opinion: biology is easier.
The Earth's magnetic field, although it does not deflect all charged particles; it only deflects most of them.
Go into space. Otherwise within the constraints of the earth's gravitational field no one does.
Less than 1 AU for sure!
The gravitational field of Earth - or any other object for that matter - gets weaker and weaker at larger distances, but never entirely drops to zero.
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