Quasi-circular spiral transfer ($\Delta v=\sqrt{\mu }(r_0^{-1/2}-r_1^{-1/2})$)

A weak tangential force can move a satellite through a slowly varying sequence of nearly circular Kepler orbits. For constant satellite mass, $E=-\mu m/(2r)$ and $v_c=\sqrt{\mu /r}$ give $\dot{E}=F{v}_c$, hence $F/m=\sqrt{\mu }\dot{r}/(2r^{3/2})$. Integrating the thrust magnitude for a monotone outward transfer proves the displayed formula. The integrated thrust per mass is not the vector difference of the endpoint velocities; gravity acts throughout the transfer. The approximation requires slow orbital evolution and negligible eccentricity.