= Quasi-circular spiral transfer
{title2=$\Delta v=\sqrt\mu(r_0^{-1/2}-r_1^{-1/2})$}
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 $\dot E=Fv_c$, 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.
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