Hohmann Transfer Orbit Calculator

For very large radius ratio changes (r2/r1 > 11.94), a bi-elliptic transfer using three burns can be more efficient. However, the bi-elliptic transfer takes much longer (it goes out to a very large intermediate orbit). For typical Earth orbital maneuvers and most interplanetary missions, Hohmann is essentially optimal for two burns.

The vis-viva equation gives the speed at any point in an orbit: v^2 = GM*(2/r - 1/a), where r is the current distance from the central body and a is the semi-major axis. For a circular orbit (a = r), this simplifies to v = sqrt(GM/r). For the periapsis and apoapsis of the transfer ellipse, r = r1 or r2 and a = (r1+r2)/2.

The transfer time is half the orbital period of the transfer ellipse: T/2 = pi*sqrt(a^3/GM), where a = (r1+r2)/2. This ranges from about 5 hours for LEO-to-GEO to 259 days for Earth-to-Mars. During this time, the target body must have moved to exactly the right position to be at the arrival point when the spacecraft arrives — this defines the launch window.

For an interplanetary Hohmann transfer, the target planet must be ahead of the spacecraft by the right angle at departure, so that it arrives at the destination when the spacecraft does. For Mars, the optimal phase angle is about 44 degrees (Mars must be 44 degrees ahead of Earth at departure). This alignment repeats every synodic period (~26 months for Mars).

The Hohmann transfer assumes coplanar circular orbits. If the initial and target orbits have different inclinations, a combined plane change and altitude change is needed. Performing the plane change at the point of highest velocity is most efficient, but plane changes are very expensive in delta-v. The Hohmann transfer provides a baseline; inclination corrections add significantly to the total cost.

The Oberth effect means burns at high velocity (low altitude) are most efficient. In a Hohmann transfer, the first burn at periapsis (lowest altitude, highest velocity) benefits from the Oberth effect. Missions that use a low perigee for departure (lunar and interplanetary) exploit this by firing at periapsis rather than from a higher circular parking orbit.

GTO (Geostationary Transfer Orbit) is the transfer ellipse used to move from LEO to GEO. Its periapsis is in LEO and apoapsis is at GEO altitude. The launch vehicle delivers the satellite to GTO; the satellite's own apogee kick motor fires the second burn to circularize at GEO. This is the standard commercial satellite launch profile.

The Hohmann calculation assumes circular coplanar orbits and instantaneous (impulsive) burns. Real planets orbit in slightly elliptical, inclined orbits. Real burns take minutes to hours. Real missions use patched-conic approximation or full numerical integration to compute accurate trajectories. The Hohmann result is an excellent first estimate for mission planning.

A porkchop plot is a contour map showing total delta-v (or C3 launch energy) versus departure date and arrival date for an interplanetary mission. It reveals the optimal launch window (minimum delta-v) and how much extra propellant is needed for off-optimal departure dates. The shape of the minimum-delta-v region resembles a porkchop, giving the plot its name.