The cheapest way between two orbits, and why it is so slow

Moving a spacecraft from one circular orbit to a larger one sounds like it should mean pointing outward and thrusting. It does not. Thrusting outward is nearly useless; the manoeuvre is performed by thrusting forwards, twice, half an orbit apart, with a long coast in between.

The reason is that in orbit, energy and altitude are the same currency. Adding speed at one point raises the opposite side of the orbit, and everything about orbital manoeuvring follows from taking that seriously.

Two burns, and why exactly two #

A single impulsive burn cannot move a spacecraft between two circular orbits, and the reason is geometric. A burn changes the velocity at the point where it happens but not the position, so the new orbit must still pass through that point. Two circles that do not intersect share no point, so no single orbit can be tangent to both after one burn.

What one burn can do is stretch the circle into an ellipse whose near end is the old orbit and whose far end reaches the new one. That takes care of getting there. Arriving is a separate problem: at the far end the spacecraft is at the right altitude and moving too slowly for a circular orbit at that radius, so without a second burn it falls straight back where it came from.

Hence two burns. The first raises the apoapsis, the second circularises. Both are prograde — along the direction of motion — because both are adding energy.

Both costs come from the vis-viva relation,

$$v^2 = GM\left(\frac{2}{r} - \frac{1}{a}\right),$$

evaluated at the same radius on the two orbits and subtracted. There is nothing more to the calculation than that.

The paradox that makes it hard to fly #

The most counterintuitive fact in the subject is that speeding up makes a spacecraft slower.

Fire prograde and the orbit rises. A higher orbit has a longer period and a lower average speed, so after half a revolution the spacecraft is behind where it would have been. To catch something ahead in the same orbit, the correct manoeuvre is to fire retrograde — slow down, drop into a lower and faster orbit, gain ground, then raise back up.

This defeated the Gemini 4 rendezvous attempt in 1965. The crew tried to close on a spent booster by pointing at it and thrusting, watched it recede, thrust harder, and watched it recede faster. They expended nearly half the mission’s fuel and gave up. The successful technique — approach from below, in a lower and faster orbit — was flown three months later on Gemini 6A.

The underlying statement is that in orbit there is no separate control over speed and altitude. There is one quantity, the orbital energy, and it appears as the semi-major axis in the vis-viva relation.

The cost, and why it is a minimum #

For a transfer from radius $r_1$ to $r_2$, the two burns in units of the initial circular speed come to a total that depends on the ratio alone.

The Hohmann transfer is the minimum-fuel solution for two impulsive burns between coplanar circular orbits, and Hohmann proved it in 1925 — twelve years before the first liquid-fuelled rocket reached a kilometre.

It is a minimum with a caveat that is genuinely surprising. Above a ratio of about 11.94, a bi-elliptic transfer beats it: burn to an ellipse reaching far beyond the target, circularise briefly out there, then drop back down. Three burns and an enormous detour, and it costs less, because the plane-change and circularisation are cheapest where the orbital speed is lowest. Below the threshold the Hohmann wins; above it the detour does. That such a threshold exists at all is the sort of result that only falls out of doing the algebra.

The threshold is actually two numbers, and the gap between them is instructive. Below a ratio of 11.94 no bi-elliptic transfer can beat the Hohmann, whatever intermediate apoapsis is chosen. Above 15.58 every bi-elliptic transfer beats it, however modest the detour. Between the two the answer depends on how far out the intermediate apoapsis is placed: a small detour loses and a large one wins. The band exists because two effects are trading against each other — a higher intermediate apoapsis makes the circularisation burn cheaper, since orbital speed falls as $r^{-1/2}$, but makes the first burn more expensive, and the two only cross at a ratio large enough that the far end of the detour is nearly free.

It is a real effect and it is almost never used. Geostationary transfer sits at a ratio of about 6.6, well below the threshold; interplanetary transfers are not between circular orbits about the same primary in any useful sense. The bi-elliptic transfer is mostly of interest as a demonstration that “minimum” in this subject always carries a domain attached to it.

What was actually flown #

The relation between the algebra and a real mission is closer than it usually is in this subject, which makes it worth checking against numbers that were paid for in propellant.

Low Earth orbit sits at about 6,570 km from the Earth’s centre, where circular speed is 7.79 km/s. Geostationary orbit sits at 42,164 km, where it is 3.07 km/s. The ratio is 6.42, and the two burns come out at 2.42 and 1.47 km/s, for 3.89 km/s total. That is the standard geostationary transfer budget, and it is the number every commercial satellite operator plans against. The measured figure for an actual flight differs from it by the plane change out of the launch site’s inclination — which is why launching from near the equator is worth money, and why Kourou at 5.2° north charges a premium over Cape Canaveral at 28.5°.

The Mars case is the one where the model’s limits show. A Hohmann transfer from Earth’s orbit to Mars’s needs 2.94 km/s of heliocentric velocity change, which sounds modest. It is not what a mission pays. The spacecraft has to escape Earth first, and the departure burn is made from low orbit where the vehicle is already moving at 7.8 km/s, so the burn is worth far more than its own magnitude — about 3.6 km/s from low Earth orbit buys both the escape and the heliocentric change. That leverage is the Oberth effect, and it is the reason departure burns are made as deep in the gravity well as possible.

The Oberth effect is worth stating as arithmetic, because it is one of the few places in orbital mechanics where the intuitive account is not merely incomplete but backwards. Kinetic energy goes as $v^2$, so a fixed $\Delta v$ applied at speed $v$ adds $v\,\Delta v$ of energy per unit mass — proportional to how fast the vehicle was already going. A burn at periapsis of a highly eccentric orbit, where the vehicle is moving fastest, therefore buys more energy than the identical burn at apoapsis. New Horizons crossed Jupiter’s orbit having spent less propellant than a direct Hohmann to Jupiter would have needed, and Parker Solar Probe reaches the Sun by repeatedly removing energy at the point where removal is worth the most.

The waiting #

The efficiency is bought with time, and the bill is large.

The coast is half the period of the transfer ellipse, and the third law gives it immediately from the semi-major axis $a = (r_1+r_2)/2$. Earth to Mars: 8.5 months. Earth to Jupiter: 2.7 years. Earth to Neptune: about 30 years, which is why nothing has done it that way.

Worse, the target has to be in the right place. The spacecraft arrives at a specific point half an orbit later, and the destination planet must arrive there at the same moment. That condition is satisfied only when the two planets start at a particular relative angle, and since both are moving, the alignment recurs at the synodic period:

$$\frac{1}{T_{\text{syn}}} = \left|\frac{1}{T_1} - \frac{1}{T_2}\right|.$$

For Earth and Mars that is 25.6 months, which is why Mars missions launch in clusters and then nothing goes for two years. For Earth and Venus it is 19.2 months. For a distant target whose period is much longer, the synodic period approaches Earth’s year, so outer-planet windows come round annually — but the alignments that permit useful gravity assists are much rarer. The Voyager grand tour needed an arrangement of the four outer planets that recurs every 175 years.

What the model leaves out #

Real missions are not two impulses between coplanar circles, and each departure from that has a cost.

Plane changes. Orbits are rarely coplanar, and rotating an orbital plane is expensive: changing direction by an angle $\theta$ costs about $2v\sin(\theta/2)$, which for a 30° change is half the orbital speed. Since the cost is proportional to $v$, plane changes are made where the spacecraft is moving slowest — at apoapsis — and combined with another burn wherever possible.

Finite burns. The analysis assumes an instantaneous velocity change. A real burn lasts minutes and the spacecraft moves during it, so some thrust is wasted at the wrong attitude. The loss is small for chemical rockets and total for electric ones, whose thrust is so low that the trajectory is a slow spiral instead.

Atmosphere. Aerobraking uses a planet’s atmosphere to do the circularisation for free, at the cost of many passes and considerable nerve. Mars Global Surveyor saved a large fraction of its propellant that way.

Gravity is not just the primary’s. Near a planet the trajectory is dominated by that planet, and interplanetary transfers are stitched together from separate two-body arcs — the patched-conic approximation, which is not rigorous and works.

The waiting, priced #

The coast time is not an inconvenience attached to the transfer. It is the transfer’s other half, and it comes from the same law. Eight and a half months to Mars, 2.7 years to Jupiter, about thirty to Neptune. Those are consequences of a three-halves power and nothing else, and no improvement in propulsion changes them for a minimum-energy trajectory.

That is the whole argument for a gravity assist, which buys speed with geometry rather than propellant and cuts the outer-solar-system numbers by factors of two or three. It is also the argument for accepting a higher-energy transfer when time matters: the minimum-fuel route is only optimal if fuel is the thing being minimised.

The trade has a shape worth knowing. Spending slightly more than the Hohmann cost buys a disproportionate amount of time at first, because the transfer ellipse gains eccentricity quickly for a small extra burn and the arrival happens further round a shorter arc. Beyond a certain point the return collapses: halving a nine-month cruise costs several kilometres per second, and getting to a Mars transit under four months requires leaving on a hyperbolic heliocentric trajectory, which no chemical stage of reasonable mass can do. The curve of time against fuel is steep at the cheap end and nearly flat at the expensive one, which is why almost every uncrewed mission flies within a few hundred metres per second of the minimum and every crewed proposal argues about the rest.

The generalisation: transfers are all one problem #

The Hohmann transfer looks like a special case, and it is — but of a problem general enough that recognising it changes what the special case means.

The general question is Lambert’s: given a departure point, an arrival point and a flight time, find the conic that connects them. It has a solution, it is unique for a given number of revolutions and a given direction of travel, and it is solved numerically in microseconds. Every real trajectory design is a search over Lambert solutions, with departure date on one axis and arrival date on the other and the required departure energy as the value at each point. Plotted as contours, that surface is the porkchop plot every mission planner works from, and the Hohmann transfer is one point on it — the bottom of the deepest basin, when the two orbits happen to be circular and coplanar.

Seeing it that way explains the launch window without any extra machinery. The basin is a basin: costs rise in every direction from the minimum, and a window is nothing more than the contour at whatever departure energy the launch vehicle can deliver. A more capable rocket does not move the minimum; it draws a wider contour around it, and the window gets longer. That is the actual relationship between rocket performance and schedule, and it is invisible in a diagram of two circles and an ellipse.

The second burn nobody makes #

The transfer above has two burns and a whole class of missions pays for only the first.

A flyby does not circularise. A spacecraft sent past a planet arrives on a hyperbola, takes its data during the encounter, and leaves — so the arrival burn, which for an orbiter is the larger part of the cost at the far end, is simply not made.

The difference is not marginal. Arriving at Jupiter from a minimum-energy transfer, the excess speed relative to the planet is several kilometres a second, and capturing into even a very eccentric orbit costs of order one. Capturing into a close circular one costs several more. A flyby pays none of it.

That is why the outer solar system was reconnoitred by flybys and orbited afterwards, and why the orbiters that followed were much larger vehicles or arrived by longer and cleverer routes. The two Voyagers flew past four planets between them; the first Jupiter orbiter arrived seventeen years after the first Jupiter flyby and needed a gravity-assist sequence to afford the capture.

There is a third category between them. A lander pays the arrival cost and then some, unless the destination has an atmosphere — in which case the atmosphere does the braking and the vehicle pays for a heat shield instead of propellant. That substitution is the reason Mars has been landed on many times and Mercury has not been landed on at all.

The cost of a mission is decided less by where it goes than by what it does on arrival, and the transfer this essay prices is the part all three have in common.

And the extreme ratio, where the two-burn transfer stops being the cheapest option.

Where the model stops #

Impulsive burns. Above.

Coplanar circular orbits. Real orbits are elliptical and inclined, and the general transfer between two arbitrary orbits is Lambert’s problem, solved numerically.

Two bodies. Low-energy transfers exploiting the Lagrange points can beat the Hohmann cost substantially, by using the three-body dynamics that the two-body analysis cannot represent. They take years, and the Genesis and GRAIL missions flew them.

Minimum fuel is the objective. It usually is not. Crewed missions trade fuel for time because radiation exposure and consumables scale with duration, and a faster trajectory that costs more may be the cheaper mission overall.

The figures have a specific limitation worth stating: they show the transfer in the frame of the primary, where the target orbit is a circle waiting to be reached. In that frame the manoeuvre looks like a shape problem. The actual difficulty is a timing problem — arriving at the same place at the same moment as a moving target — and no static diagram can show it. The launch window is invisible in every Hohmann diagram ever drawn, and it is the constraint that determines when missions fly.

A last note on the word “minimum”, which appears throughout and always with a domain attached. The Hohmann transfer is minimal among two-impulse transfers between coplanar circular orbits below a ratio of 11.94. Change any clause and a different route wins: three impulses above that ratio, a low-energy transfer through the Lagrange points if time is free, an aerobraking arrival if the destination has air. Nothing here is the cheapest way to get anywhere in general, and the phrase is always short for a longer sentence.

The transfer’s cost depends on one number, the ratio of the two radii, and it is worth drawing at three values that span what has actually been flown.

The ladder from here #

Later rungs: the vis-viva relation derived from energy. The Hohmann cost as a function of ratio, and the bi-elliptic crossover. Plane changes and combined manoeuvres. Lambert’s problem. Synodic periods and porkchop plots. The oberth effect, and why burns deep in a gravity well are worth more. Low-thrust spirals and electric propulsion. Aerobraking. Weak stability boundary transfers. And the rocket equation, which says what any of this costs in propellant rather than in velocity.

Hohmann worked the transfer out as a hobby while employed as an architect, and published it in a book on the attainability of celestial bodies. Rockets capable of using it did not exist for another thirty-two years.