A satellite of mass m moves under the attractive inverse square field -(mγ /r2)r^ and is also subject to the linear resistance force -mKv, where K is a positive constant. Show that the governing equations of motion can be reduced to the form
r·· + Kr· + (γ/r2) - (L02e-2Kt/r3) = 0,
r2θ· = L0e-Kt,
where L0 is a constant which will be assumed to be positive. Suppose now that the effect of resistance is slight and that the satellite is executing a ‘circular' orbit of slowly changing radius.
By neglecting the terms in r and r, find an approximate solution for the time variation of r and θ in such an orbit. Deduce that small resistance causes the circular orbit to contract slowly, but that the satellite speeds up!