Hohmann Transfer Calculator

A Hohmann transfer moves a spacecraft between two coplanar circular orbits along an ellipse tangent to both, with one burn at each tangent point. The first burn, applied at the inner radius, raises apoapsis out to the target radius; the second, applied on arrival at apoapsis, raises periapsis to circularize. Every speed comes from the vis-viva equation, v equal to the square root of mu times the quantity 2 over r minus 1 over a, where mu is the standard gravitational parameter, 398,600 km cubed per second squared for Earth and about 1.327 times 10 to the 11th for the Sun. The transfer ellipse has a semi-major axis equal to the mean of the two radii and an eccentricity equal to their difference divided by their sum, and the flight time is exactly half the period of that ellipse. From a 6,678km low orbit to geostationary at 42,164km the two burns total roughly 3.9 km/s over about 5.3 hours; a heliocentric Earth to Mars transfer costs roughly 5.6 km/s and takes about 259 days. The maneuver is the minimum-energy two-impulse solution only inside its assumptions: coplanar circular orbits, instantaneous burns, and a single gravitating body. Real missions pay extra for plane change, which from a 28.5-degree inclination adds well over 1 km/s at geostationary arrival, and they use patched-conic escape and capture hyperbolas at each planet. Beyond a radius ratio of about 11.94 a three-burn bi-elliptic transfer becomes cheaper, and interplanetary departures are further constrained by synodic launch windows, about 26 months for Mars.

Walter Hohmann was a municipal building surveyor in Essen, Germany, who studied spaceflight trajectories in his own time. He published the two-impulse tangential transfer in 1925 in a slim book titled Die Erreichbarkeit der Himmelskoerper, translated as The Attainability of Celestial Bodies, computing departure and arrival energies for the planets decades before any hardware could act on them. Hohmann joined the Verein fuer Raumschiffahrt, the German rocket society whose membership included Hermann Oberth and Wernher von Braun, and his transfer quickly became the standard reference maneuver of early astronautics. He died in 1945. The bi-elliptic alternative that beats the Hohmann transfer at large radius ratios was analysed by Ary Sternfeld, a Polish-born engineer who worked in France and later the Soviet Union, during the 1930s. Practical use of the Hohmann geometry began with the first synchronous-orbit missions: Syncom 2 in 1963 and Syncom 3 in 1964 reached synchronous altitude by way of a transfer ellipse and an apogee kick motor, a pattern still followed by nearly every commercial communications satellite launched today. Mariner 4 flew a near-Hohmann heliocentric trajectory to Mars across 1964 and 1965. Later missions traded some of that efficiency for shorter flight times or for gravity assists, but the Hohmann transfer remains the baseline against which every alternative trajectory is measured.