= Hohmann transfer impulse fractions
{c}
{title2=$\Delta v_1/v_0=\sqrt{2\eta/(1+\eta)}-1$}
For a <Hohmann transfer orbit> from radius $r_0$ to $\eta r_0$, with $\eta>1$ and initial circular speed $v_0=\sqrt{GM/r_0}$, conservation of <orbital energy> and <angular momentum> gives transfer endpoint speeds $v_p/v_0=\sqrt{2\eta/(1+\eta)}$ and $v_a/v_0=\sqrt{2/[\eta(1+\eta)]}$. The required prograde impulses are
$$
\frac{\Delta v_1}{v_0}=\sqrt{\frac{2\eta}{1+\eta}}-1,\qquad
\frac{\Delta v_2}{v_0}=\frac1{\sqrt\eta}-\sqrt{\frac2{\eta(1+\eta)}}.
$$
If the spacecraft mass is constant and fuel use is proportional to impulse magnitude, these also give the fuel fractions relative to a budget with total speed change $v_0$. Their sum is a sum of magnitudes, even though the burns occur at different positions and directions.
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