Reading about the Hill Sphere I notice that "the region of stability for retrograde orbits at a large distance from the primary, is larger than the region for prograde orbits at a large distance from the primary," but this is not explained. The equations on this page do not seem to include a rotational component.
Why do prograde and retrograde orbits have different regions of stability, and how significant is this effect?
To understand this, you have to understand how the spin affects the gravity. The spin creates a sling shot effect of pushing an object in the direction of the spin. With that in mind, in a retrograde orbit, there are two forces pushing in opposite directions keeping an object in a purgatory state like. Essentially it's like a game of tug of war between the masses. This creates a larger space of stability due to equal and oppose forces being exerted. In prograde, it's like having two tires spinning in the same direction touching each other and putting a marble between the tires. The marble would then be shot out due to the two forces acting on it in the same direction. The region of stability decreases due to the two forces pushing in the same direction. Thankyou Dr. Bill for suggesting that. Sorry I didn't completely elaborate on the "forces" involved in my explanation. The forces are gravity and magnetism, so it's not like water on the surface of a spinning ball. Here's why, relative to the distance between the objects, if the masses are large enough combined with a strong enough magnetic field, then there will be an observable affect of the range of stability. The Earth spins as a result of a constant electric current, and a constant magnetic field as well as the changing Angle in-between. To answer the original question, the spin affects the magnetic field directly. To demonstrate this affect, grab a pair of magnets and hold one in each hand at your fingertips. Then spin them around each other within the magnetic field. Next simulate retrograde and prograde rotation and see what happens. It will be easier to spin them in opposite directions as you twirl them around each other.
