Why do satellites orbit around the centre of a planet? Why can satellites not  orbit  around the North or South Pole, instead of  orbiting about the centre of the earth? 
 A: The sum of the gravitational forces on a satellite from all the atoms in the Earth is a net force toward the center of the Earth. (This is a consequence of spherical symmetry. It can also be shown by using integral calculus.) Thus any satellite’s orbit must accelerate toward the center and not toward, say, a pole.
A: Here is a diagram showing the orbit of a satellite around the North Pole.  
 
The gravitational force $\vec F$ can be split into two components $\vec {F'}$ and $\vec {F''}$ such that $\vec {F}=\vec {F'} + \vec {F''}$.
The component in the plane of the orbit $\vec {F'}$ provides the centripetal acceleration around the North Pole $r\omega^2$.  
However there is still the component $\vec {F''}$ to deal with as otherwise there will also be an acceleration of the satellite in the direction of this force.
One would need to have the satellite equipped with a rocket engine continuously providing a force equal in magnitude but opposite in direction to force $\vec {F''}$ to keep the satellite in the required orbit.
This is not a feasible proposition.
A: If we ignore the cases of elliptical orbits, the earth gives the satellite a gravitational attraction towards its centre, which is perpendicular to the velocity of the satellite. In your question, such a motion requires a force towards the poles of our earth, which is impossible to satisfy because a component force of the gravity would pull the satellite downwards and disrupt its original motion.
