Randall D. Knight — Newton's Theory of Gravity: Problem 68

A satellite in a circular orbit of radius \( r \) has period \( T \). A satellite in a nearby orbit with radius \( r + \Delta r \), where \( \Delta r \ll r \), has the very slightly different period \( T + \Delta T \). a. Show that \[ \frac{\Delta T}{T} = \frac{3}{2} \frac{\Delta r}{r} \] b. Two earth satellites are in parallel orbits with radii 6700 km and 6701 km. One day they pass each other, 1 km apart, along a line radially outward from the earth. How long will it be until they are again 1 km apart?