Hohmann Transfer Calculator

The cheapest way between two circular orbits is half an ellipse that kisses both. Enter the two orbit radii (in AU, for craft around the Sun), and this calculator returns the two engine burns the transfer needs and the months of patience it demands in exchange.

Half an Ellipse, Two Burns, One Bargain

Walter Hohmann, a German engineer who calculated in his spare time, published the idea in 1925: to move between circular orbits, burn once to stretch your orbit into an ellipse whose far end just touches the target circle, coast half a revolution, and burn again to circularise at the new radius. The physics is the vis-viva equation — speed at any point of an orbit from its semi-major axis — applied four times: your circular speed, the ellipse's speed at departure, the ellipse's speed at arrival, and the target's circular speed. The two differences are the burns, and every probe that has cruised to Mars or Venus has flown essentially this ellipse, because no cheaper two-impulse route between coplanar circles exists; the cost of the elegance is time — half an orbital period of the transfer ellipse, spent coasting in the dark.

Worked Example: Earth to Mars, the 259-Day Ellipse

The defaults are the canonical voyage: r₁ = 1.000 AU, r₂ = 1.524 AU. Burn 1 = 2.95 km/s (leaving Earth's orbit onto the ellipse), burn 2 = 2.65 km/s (matching Mars at arrival), total Δv = 5.60 km/s, flight time = π√(a³ ÷ μ) with a = 1.262 AU = 259 days, about eight and a half months. Those are the actual line items behind every Mars mission's budget — NASA's cruise phase lasts almost exactly this long, and the launch windows every 26 months exist because the planets must be phased so Mars arrives at the meeting point when the ellipse does (the synodic calculator in this wave prices that wait). The calculator omits the planets' own gravity wells (escaping Earth and being captured by Mars are separate line items, roughly 3.6 and 2.6 km/s in practice) — heliocentric transfer only, stated plainly, as Hohmann framed it.

Delta-v: The Currency of Spaceflight

Spacecraft do not buy distance; they buy Δv, and the rocket equation collects payment exponentially — every extra km/s multiplies the fuel fraction, which is why mission designers speak of Δv budgets the way accountants speak of cash. The Hohmann transfer is the minimum entry in that budget for its route, and its numbers set the feasibility baseline for everything louder: a bi-elliptic transfer can undercut it for radius ratios above about 12 (at the price of far longer flight), gravity assists steal Δv from planets (the Voyager and Cassini trick), and low-thrust ion craft ignore Hohmann entirely, spiralling instead — slower, cheaper per kilogram, and un-flyable by this page's impulsive-burn model. The calculator therefore prices the benchmark: the route against which every cleverer trajectory must justify itself.

The Grand Tour's Debt

Every outer-planet mission is a Hohmann transfer with planets substituted for patience: the Voyager grand tour exploited a 176-year alignment to chain gravity assists, but the legs between planets remained this page's ellipses, their timing computed from the same vis-viva ledger. Even the humble Earth–Moon hop wears the shape: Apollo's translunar injection stretched its parking orbit into an ellipse whose far end met the Moon as it arrived — a Hohmann transfer with a moving, gravitating target and a figure-eight correction drawn by hand at Mission Control. The calculator prices the clean two-body case; the history of flight is that case, elaborated by geniuses who started from Hohmann's spare-time arithmetic and added the universe's complications one term at a time.

When Not to Hohmann

The benchmark has honest limits, and mission design is largely the art of knowing them: for radius ratios beyond about twelve, a bi-elliptic transfer (two half-ellipses via a very distant apoapsis) undercuts Hohmann's Δv while multiplying the flight time beyond most planners' patience; for cargo that can wait years, low-thrust spiral trajectories trade the impulsive burns for months of whisper-thrust at a fraction of the propellant; and for crewed flight, where time is life-support mass, faster non-Hohmann transfers burn extra Δv deliberately — the calculator's minimum is a price floor, not an instruction. Knowing the floor is what makes the departures from it rational rather than merely expensive, which is why this page belongs in the planner's kit even when the mission flies otherwise.

The Oberth Dividend

A refinement every mission planner applies to this page's raw burns: a burn executed deep in a gravity well (at periapsis, where the craft moves fastest) buys more orbital energy per unit of Δv than the same burn in the shallows — the Oberth effect, a consequence of the kinetic energy's v² bookkeeping, since the exhaust's discarded energy depends on the craft's speed at firing. Hohmann's two burns are already placed at the ellipse's apsides where the effect helps most, one reason the old transfer stays competitive against cleverer geometries. The calculator prices the ideal impulses; Oberth is the discount the real trajectory quietly applies on top.

Related Tools

More from the Physics Calculators hub: the Synodic Period Calculator — when the window for this transfer opens — and the Circular Motion Calculator, for the circular speed both burns start from.

Frequently Asked Questions

Why does the transfer take exactly half an ellipse?

Because the ellipse's two apsides are the touching points: departure at one end of the major axis, arrival at the other, and the craft traverses the half of the ellipse between them. The flight time is thus half the period of an orbit with semi-major axis (r₁ + r₂) ÷ 2 — Kepler's third law in its travelling salesman's form. A full ellipse would bring the craft back to the departure circle, which is the geometry of a rendezvous that failed: several comets ride full Hohmann-like ellipses and meet nobody, every few years, forever.

What if the orbits are not coplanar?

Then a plane-change burn joins the invoice, and it is brutally priced: tilting a velocity vector of speed v by an angle θ costs 2v sin(θ ÷ 2), which at orbital speeds dwarfs the Hohmann burns for even a few degrees — the reason launch windows also wait for the planets' orbital planes to be favourably aligned, and why polar-orbit missions launch from sites and at times the map makes non-obvious. This calculator assumes coplanar circles, the textbook case and, for the ecliptic-hugging planets, a fair approximation of the real invoice's main lines.

Can I use it for Earth-orbit satellites?

The same mathematics with Earth's own μ and radii in kilometres — the GEO transfer is a Hohmann from low orbit to the geostationary circle, and thousands of satellites have ridden it (with the second burn doubling as the circularisation at apogee). This page is calibrated for the Sun (radii in AU), so the satellite case needs the generic formula rather than these inputs — the method, the two burns and the half-ellipse are identical, only the central mass and the units change chairs. The honesty of a calculator is partly in its stated jurisdiction; this one's is heliocentric.

Why is burn 1 bigger than burn 2, going outward?

It usually is not by much, and for Earth–Mars it is (2.95 vs 2.65) because the ellipse's departure point is its perihelion, where the craft must gain the most speed relative to the local circular value, while at the aphelion end the circular speed of the target lies closer to the ellipse's lazy arrival speed. The vis-viva equation prices both without sentiment: speed is greatest where the orbit is deepest in the gravity well. For transfers to Venus (inward), the signs reverse — both burns become braking burns, and the first is again the larger.

What is a launch window, in these terms?

The target planet must reach the arrival point at the same moment the craft does — so departure must happen when the target is ahead of the arrival point by exactly the angle it will traverse during the flight time. That phasing recurs with the synodic period (for Mars, every 780 days), which is why missions cluster in frantic seasons and then the pad sits quiet for two years. The calculator gives the flight time that the phasing is built from; the synodic page in this same wave gives the recurrence — between them, a mission's calendar is fully stated.

How long is the flight to Jupiter on this scheme?

Set r₂ = 5.203 and the calculator returns the patient answer: the transfer ellipse has a = 3.10 AU, and its half-period is about 2.73 years — the actual cruise time of Galileo and Juno's inner-system legs' geometry, before gravity assists rewrote their itineraries. Saturn pushes it past six years. The numbers explain an entire philosophy of outer-planet exploration: instruments must be built to survive the answer this page gives, and careers are planned around it — a mission scientist may shepherd two launches in a working lifetime, each preceded by a Hohmann half-ellipse of years.

Who was Walter Hohmann?

Walter Hohmann (1880–1945) was a Munich construction engineer who, bedridden during the First World War, worked out the mathematics of interplanetary flight as a private obsession and published it in 1925 in Die Erreichbarkeit der Himmelskörper — "The Attainability of the Heavenly Bodies" — a book that seeded the German rocket societies and, through them, much of the century's spaceflight. NASA translated and studied it a generation later. The transfer that bears his name has since carried probes to every planet — the spare-time ellipse of a building engineer, flown by nations.