Astronomy & Celestial Mechanics
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Spaceflight

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

Two burns and a long coast is the least fuel that will move a spacecraft between two circular orbits. It is also, for anything beyond the Moon, an unreasonably long wait.
10 min read 6 figures Everything is a conicThe sky as a clock

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.

A Hohmann transfer, 2.6 to 1 in radiusTwo circular orbits and the ellipse that touches both. The first burn raises the far point to the outer orbit; the second, half an orbit later, circularises. The costs are computed from the vis-viva relation.burn 1: +0.202burn 2: +0.158total 0.360 — and no way to spend lesscoast: half an ellipse, 1.21 of an inner-orbit yearspeeds in units of the inner circular speed
Fig. 1 A transfer between two circular orbits. The first burn raises the far point of the orbit to the outer radius; the second, half an orbit later, circularises there. Both costs are computed from the vis-viva relation.

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,

v2=GM(2r−1a),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.

Speed against distance, for orbits of the same periodOrbital speed against distance from the primary. Every orbit with the same semi-major axis follows the same curve; the eccentricity decides only which stretch of it the body uses.00.511.50123distance from the primary (a = 1)e = 0e = 0.4e = 0.8circular speed at r = a
Fig. 2 Speed against distance for orbits sharing a semi-major axis. A burn moves the spacecraft from one such curve to another at fixed radius, and the size of the jump is the cost.

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 r1r_1 to r2r_2, the two burns in units of the initial circular speed come to a total that depends on the ratio alone.

A Hohmann transfer, 1.6 to 1 in radiusTwo circular orbits and the ellipse that touches both. The first burn raises the far point to the outer orbit; the second, half an orbit later, circularises. The costs are computed from the vis-viva relation.burn 1: +0.109burn 2: +0.097total 0.207 — and no way to spend lesscoast: half an ellipse, 0.74 of an inner-orbit yearspeeds in units of the inner circular speed
Fig. 3 A modest transfer, ratio 1.6 to 1. Both burns are small, the transfer ellipse barely differs from either circle, and the coast is short.
A Hohmann transfer, 6 to 1 in radiusTwo circular orbits and the ellipse that touches both. The first burn raises the far point to the outer orbit; the second, half an orbit later, circularises. The costs are computed from the vis-viva relation.burn 1: +0.309burn 2: +0.190total 0.499 — and no way to spend lesscoast: half an ellipse, 3.27 of an inner-orbit yearspeeds in units of the inner circular speed
Fig. 4 A large transfer, ratio 6 to 1. The first burn is much bigger, the second smaller, and the coast is a long ellipse — the outer end of which is where the arrival happens.

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 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=(r1+r2)/2a = (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:

1Tsyn=∣1T1−1T2∣.\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 θ\theta costs about 2vsin⁡(θ/2)2v\sin(\theta/2), which for a 30° change is half the orbital speed. Since the cost is proportional to vv, 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.

Circular and escape speedOrbital and escape speed against distance, in units of the circular speed at the surface. The escape curve is the circular one multiplied by the square root of two, at every distance without exception.2468101200.511.5distance, in body radiiescape speedcircular speedthe surfacea high orbitgeostationarythe gap is always a factor of √2
Fig. 5 Circular and escape speed against distance. A transfer moves between the two curves without crossing the upper one; a mission that does cross it is on an entirely different kind of trajectory.

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.

Period against size for the planets, around the SunOrbital period against semi-major axis on logarithmic axes. The line has slope exactly three-halves — the harmonic law — and the measured bodies sit on it.0.321.03.210320.100.321.03.21032100316semi-major axis (AU)MercuryVenusEarthMarsJupiterSaturnUranusNeptuneslope 3/2 — P² ∝ a³period (years)
Fig. 6 Period against semi-major axis. The transfer ellipse’s semi-major axis is the average of the two orbital radii, so the harmonic law returns the flight time in a single step.

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.

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.

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.

Where this goes next

0.321.03.210320.100.321.03.21032100316semi-major axis (AU)MercuryVenusEarthMarsJupiterSaturnUranusNeptuneslope 3/2 — P² ∝ a³period (years) The law that links period to size, and weighs everything Kepler found that the square of the period goes as the cube of the orbit. Newton found the constant of proportionality, and that constant is a mass — which is how every mass in astronomy has been obtained since. 2468101200.511.5distance, in body radiiescape speedcircular speedthe surfacea high orbitgeostationarythe gap is always a factor of √2 The speed that does not come back, and the √2 that separates it Escape speed is exactly the square root of two times circular speed, at every distance from every body. Being in orbit is already 71% of the way to leaving. the planet's velocityinout70°speed before: 0.505speed after: 0.661gained 0.157, and burned nothingthe circle: constant speed in the planet's frame Stealing speed from a planet, which does not notice A flyby cannot change a spacecraft's speed relative to the planet. It changes its direction — and adding the planet's own motion back turns that into free velocity.