Hohmann transfer orbit

A Hohmann transfer uses two tangential impulses and one half of an elliptic Kepler orbit to move between two coplanar circular orbits.

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.

An outward transfer from radius $R$ to $\alpha R$ arrives with speed $v_a/v_c=\sqrt{2/(1+\alpha )}$, where $v_c$ is the desired outer circular speed. Reversing the same circularization impulse leaves speed ratio $v_w/v_c=2\sqrt{2/(1+\alpha )}-1$, possibly signed. Its square $q$ determines the new apocentre orbit: $e=1-q$ and $r_p=\alpha Rq/(2-q)$. The orbit intersects a spherical body of radius $R_b$ if $r_p<R_b$. A negative signed speed means retrograde motion, while $q=0$ is the degenerate radial-fall limit.

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 areIf 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.

A Hohmann transfer orbit between coplanar circular Kepler orbits of radii $r_1,r_2$ traverses half an elliptic orbit of semi-major axis $a=(r_1+r_2)/2$. Kepler's third law therefore gives $t=\pi \sqrt{a^3/(GM)}$. Relative to the initial orbital period $T_1$, $t=(T_1/2)[(r_1+r_2)/(2r_1){]}^{3/2}$. Radii must be measured from the attracting centre, rather than from its surface.