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For $u=\sqrt{r_1/r_0}>1$, the sum of the two Hohmann transfer impulse magnitudes minus the integrated thrust per mass in a quasi-circular spiral transfer isIt is negative because $(u+1{)}^2<2(u^2+1)$ is equivalent to $(u-1{)}^2>0$. Thus the Hohmann transfer uses less ideal total impulse for any strictly outward pair of coplanar circular orbits. This comparison does not optimize over all other transfer families.

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.