The Hohmann Transfer

The two-impulse transfer between coplanar circular orbits

Essen, Germany, 1925

Walter Hohmann designs buildings. He is the city architect of Essen, responsible for municipal construction in an industrial city still recovering from a lost war. In his evenings he works out how to move between planets. In 1925 he publishes Die Erreichbarkeit der Himmelskörper — "The Attainability of Celestial Bodies" — a slim book containing, among other things, the minimum-energy transfer between two circular orbits. He proves it is optimal. He tabulates the costs for every planet. Nothing has ever been in space. The V-2 is seventeen years away. The first satellite is thirty-two years away. Hohmann will die in 1945, during the bombing of Essen, twelve years before Sputnik. And the transfer he worked out in his spare time is still, a century later, the first calculation anyone runs when planning a mission — and the one that actually gets flown, most of the time, because he was right.

Watch first · presented by Shoaib Aziz

The numbers in this film are the labs' own: from a 400 km orbit to GEO the transfer ellipse has a = 24,467.5 km, burn 1 is 2.40 km/s (7.67 → 10.07), burn 2 is 1.46 km/s (1.62 → 3.07), 3.86 km/s in all, 5.29 hours; GPS costs 3.42 km/s, a 400 km climb 217 m/s, a graveyard push 11 m/s; the cost peaks at a radius ratio of 15.58; and 2 t to GEO needs 5,083 kg of propellant chemically but 681 kg on a spiral that costs 19% more Δv.

The idea is almost aggressively simple. To move from a small circle to a large one, use an ellipse that touches both: perigee on the inner circle, apogee on the outer one.

Two burns, both tangential, both at an apsis.

Burn 1 at the inner orbit: speed up just enough to raise apogee out to the target radius. Coast half an ellipse, engines off, for half the transfer period. Burn 2 at apogee: speed up again to circularise, because arriving at apogee you are moving far too slowly to stay there.

Figure 6.5

The transfer ellipse tangent to both circles. Its perigee radius is r₁, its apogee radius is r₂, so its semi-major axis is the average of the two. Both burns are purely tangential — the velocity vector changes length but not direction, which is exactly what makes it optimal.

Everything follows from vis-viva. The transfer ellipse has

a_t = \frac{r_1 + r_2}{2}

and the four speeds you need are all one equation away:

\begin{aligned}v_{c1} &= \sqrt{\frac{\mu}{r_1}} & v_{t1} &= \sqrt{\mu\left(\frac{2}{r_1} - \frac{1}{a_t}\right)}\\[4pt] v_{c2} &= \sqrt{\frac{\mu}{r_2}} & v_{t2} &= \sqrt{\mu\left(\frac{2}{r_2} - \frac{1}{a_t}\right)}\end{aligned}

Circular speed at each radius, and transfer-ellipse speed at each apsis. Four numbers, one equation, no new theory required.

\Delta v_{1} = v_{t1} - v_{c1}, \qquad \Delta v_{2} = v_{c2} - v_{t2}, \qquad \Delta v_{\text{total}} = |\Delta v_{1}| + |\Delta v_{2}|

Both burns are simple subtractions because the velocity is tangential at both ends. This is the only manoeuvre in the course where Δv is a difference of magnitudes rather than a vector difference.

t_{\text{transfer}} = \frac{T_t}{2} = \pi\sqrt{\frac{a_t^3}{\mu}}

Half the period of the transfer ellipse — you fly exactly half of it. For LEO to GEO this is 5.3 hours; for Earth to Mars it is 259 days.

💡Why tangential burns at apsides are optimal

The optimality is not an accident of algebra. There are two separate reasons, and both are worth understanding because they generalise. First: no wasted direction. A burn changes energy at the rate

\dot{\varepsilon} = \vec{v}\cdot\vec{a}

— it is a dot product. Thrust perpendicular to the velocity contributes nothing to energy at all; it only rotates the velocity vector. At an apsis the flight path angle is exactly zero (Unit 2), so a tangential burn is perfectly aligned with the velocity and every newton-second goes into energy. Second: the Oberth effect. For a small burn,

\Delta\varepsilon \approx v\,\Delta v

— the energy gained is proportional to the speed you already have. Perigee is where you are fastest, so a burn there is worth more than the same burn anywhere else on the orbit. Put together: burn along the velocity, and burn where the velocity is largest. The Hohmann transfer does both, at both ends, which is why nothing beats it for moderate radius ratios.

📐Worked Example: LEO to GEO

From a 400 km circular parking orbit (

r_1 = 6771

km) to geostationary (

r_2 = 42{,}164

km). Transfer ellipse.

a_t = (6771 + 42164)/2 = \mathbf{24{,}467.5}

km. Speeds at the inner orbit.

v_{c1} = \sqrt{398600/6771} = 7.673

km/s.

v_{t1} = \sqrt{398600.44(2/6771 - 1/24467.5)} = \sqrt{101.45} = 10.072

km/s.

\Delta v_1 = 10.072 - 7.673 = \mathbf{2.400}

km/s. Speeds at the outer orbit.

v_{c2} = \sqrt{398600/42164} = 3.075

km/s.

v_{t2} = \sqrt{398600.44(2/42164 - 1/24467.5)} = \sqrt{2.616} = 1.617

km/s.

\Delta v_2 = 3.075 - 1.617 = \mathbf{1.457}

km/s. Total: 3.857 km/s. Transfer time

= \pi\sqrt{24467.5^3/398600} = 19{,}035

= \mathbf{5.29}

hours. Read the second burn again. Arriving at apogee the spacecraft is crawling at 1.62 km/s while geostationary orbit requires 3.07 km/s. Nearly half the total budget is spent not on getting to GEO but on staying there — climbing to the right altitude with insufficient speed and then having to buy the speed back. The intermediate orbit that exists between the two burns is called GTO, geostationary transfer orbit, and half the commercial satellites ever launched have sat in it waiting for their apogee kick motor to fire.

Figure 6.6

Total Hohmann Δv against the radius ratio r₂/r₁, in units of the initial circular speed. It rises, peaks near a ratio of 15.6 at about 0.536 v_c₁, and then *falls* — because for very large ratios the second burn becomes almost free. That peak is the crack that Lesson 6-3 prises open.

⚠Limitations of the impulsive Hohmann model

The Hohmann transfer is exactly optimal for a specific idealised problem. Real missions violate its assumptions constantly. Impulsive burns. The derivation assumes instantaneous velocity changes. A real burn takes minutes, during which the spacecraft moves along its orbit and the thrust direction is no longer ideal. The penalty is called gravity loss and it is typically 1–5% for a chemical stage. For a low-thrust engine, the assumption fails completely — a spiral transfer costs roughly

|v_{c2} - v_{c1}|

, which for LEO→GEO is 4.6 km/s rather than 3.85. Coplanar orbits. If the two orbits are not in the same plane you must also change the plane, and 6-4 will show that this is often the single most expensive line in the budget. Circular orbits. For eccentric orbits the tangential-at-apsides argument still applies but the geometry changes, and the optimum generally becomes a one-tangent burn rather than a clean Hohmann. None of this makes the calculation useless. It makes it the first number, the one you compute in ten seconds to find out whether the mission is plausible before spending three months on the real trajectory.

🛰Where the apogee kick actually happens

A commercial GEO satellite is usually not delivered to GEO. It is delivered to GTO — the transfer ellipse — and the second Hohmann burn is the satellite's own job. That burn, historically a solid apogee kick motor and now a liquid engine, is the single largest propellant load the satellite carries: roughly 1.5 km/s worth, which for a 3-tonne satellite is well over a tonne of propellant that must be lifted from the ground and then thrown away in one firing. Which is why so many operators now launch to a supersynchronous transfer orbit — apogee raised beyond GEO — and use electric propulsion to circularise over several months. The launcher does slightly more work, the satellite carries dramatically less propellant, and the operator trades six months of revenue for a smaller, cheaper launch slot. The underlying question — is the two-burn Hohmann really optimal? — turns out to have a more interesting answer than Hohmann himself realised.

↓But there is a problem

So the Hohmann transfer is optimal. Two burns, minimum energy, provably the cheapest way between two circular orbits. Except look again at Figure 6.6. The total cost does not increase forever with the radius ratio. It peaks, around a ratio of 15.6, and then comes back down. Something strange is happening out there: at very large ratios, going further is getting cheaper. And that raises a genuinely absurd question. If reaching a very distant orbit is cheap, and the expensive part is the region in between — what if you deliberately overshot? Fly out past your target, far past it, to somewhere enormously distant. Circularise nothing. Just coast out, make a tiny nudge where you are barely moving at all, and fall back to where you actually wanted to be. That is a longer trip, a more complicated trip, and an obviously stupid trip. Past a radius ratio of about twelve, it is also cheaper.