Past exam of the physics 1 paper of the University of Oxford / 2016 / 10 / iv / Solution

In a quasi-circular spiral transfer, a small tangential thrust $F$ supplies power $F{v}_c$ while the orbit stays approximately circular. Differentiating $E=-\mu m/(2r)$ and using $v_c=\sqrt{\mu /r}$ givesThis is the integrated thrust magnitude per unit constant mass, not the net vector change of orbital velocity.

For comparison, write $u=\sqrt{r_1/r_0}>1$. Adding the two Hohmann transfer impulses and subtracting the slow-transfer result yieldsThe bracket is negative because $(u+1{)}^2<2(u^2+1)$ for $u>1$. Thus the Hohmann transfer uses less total impulse than the slow circular spiral for these ideal coplanar transfers. The Hohmann transfer compared with a slow spiral does not establish a global optimum over every possible transfer family or propulsion constraint.