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Orbital mechanics are a direct result of the effect of gravitational force. Without gravitational force, no object would orbit any other - they all would follow their own independent straight line travel.

Also, in reference to this question's inference that gravity pulls downward (i.e. towards the earth), remember that ALL objects, regardless of size, exert gravitational force. Obviously, the larger the object, the larger the gravitational force; and, the closer the object, the more force it exerts. So, in this case, remember that things near the Earth have at least three major gravitational forces acting on them: the Earth, the Moon, and the Sun. All three gravitational forces "pull" in different directions, and exert a different amount of force (based on the exact location of the orbiting object).

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