Going too far in order to arrive cheaply

The Hohmann transfer is usually described as the cheapest way to move between two circular orbits, and the description is missing a word. It is the cheapest two-burn way.

Allow three burns and something strange becomes available. Fire once to raise the apoapsis far beyond the destination — arbitrarily far. At that distant apoapsis, where the vehicle is barely moving, fire a second time to raise the periapsis to the target radius. Then fall back and fire a third time to circularise.

Three burns instead of two, a flight time longer by orders of magnitude, and a path that goes to entirely the wrong place first. Above a radius ratio of about 11.94, it costs less total Δv than the Hohmann.

Why the Hohmann curve turns over #

The first surprise is in the Hohmann curve itself, and it is worth taking before the comparison.

With $\mu = 1$ and the inner radius 1, the two burns of a Hohmann transfer to radius $R$ cost

$$\Delta v = \left|\sqrt{\frac{2R}{1+R}} - 1\right| + \left|\frac{1}{\sqrt{R}} - \sqrt{\frac{2}{R(1+R)}}\right|.$$

That function rises from zero at $R = 1$, reaches a maximum of 0.5364 at $R = 15.58$, and then falls — approaching $\sqrt{2} - 1 = 0.4142$ as $R \to \infty$.

Going further can cost less. The reason is that the second burn changes sign in importance: for a very distant target, the first burn is nearly an escape burn and the second is a small circularisation at a place where orbital speeds are tiny. In the limit, the whole cost is the escape burn, $\sqrt 2 - 1$, and the arrival is free.

So the Hohmann cost of reaching infinity is less than the Hohmann cost of reaching fifteen times the starting radius. That is not an artefact; it is the reason the bi-elliptic works.

The bi-elliptic, and where it wins #

The bi-elliptic exploits that non-monotonicity directly. Its three burns are:

$$\Delta v_1 = \sqrt{\frac{2}{1} - \frac{1}{a_1}} - 1, \qquad a_1 = \frac{1 + r_b}{2},$$

$$\Delta v_2 = \sqrt{\frac{2}{r_b} - \frac{1}{a_2}} - \sqrt{\frac{2}{r_b} - \frac{1}{a_1}}, \qquad a_2 = \frac{r_b + R}{2},$$

$$\Delta v_3 = \left|\sqrt{\frac{1}{R}} - \sqrt{\frac{2}{R} - \frac{1}{a_2}}\right|,$$

with $r_b$ the intermediate apoapsis. Every one of them is a vis-viva speed minus another vis-viva speed at the same radius — there is nothing else in the calculation.

The middle burn is the trick. It happens at $r_b$, where the vehicle’s speed is small, and a plane or periapsis change made where the speed is small is cheap. Raising the periapsis from 1 to $R$ at an apoapsis of 200 costs almost nothing, because at that distance the difference between the two orbits’ speeds is a few per cent of a very small number.

Two thresholds come out of the comparison, and both are computable from the figure rather than quoted at it:

• Below $R = 11.94$, the Hohmann is cheaper than any bi-elliptic, however the intermediate radius is chosen.
• Above $R = 15.58$, every bi-elliptic with a large enough intermediate radius beats the Hohmann.
• Between them, it depends on $r_b$: a bi-elliptic with a sufficiently distant apoapsis wins and a modest one does not.

The trade nobody takes #

The saving is real and it is small. At $R = 20$ with an intermediate radius of 100, the bi-elliptic costs about 1.5% less than the Hohmann. At $R = 100$ it saves about 6%.

The cost is time. A Hohmann transfer takes half the period of the transfer ellipse:

$$t_H = \pi\sqrt{\frac{(1+R)^3}{8}}.$$

The bi-elliptic takes half the period of each of its two ellipses, and both have semi-major axes involving $r_b$. For the $R = 16$, $r_b = 60$ case drawn above, that is a factor of about fifteen longer. For a transfer from low Earth orbit to a lunar-distance orbit — a ratio near 60 — the Hohmann takes five days and a bi-elliptic saving 5% of the Δv takes several weeks.

That is why no flown mission has used a pure bi-elliptic transfer to reach a higher orbit. A few per cent of Δv is worth having; a factor of ten in flight time costs consumables, power, radiation exposure and operations, and for a crewed vehicle it is prohibitive — the same accounting that the rocket equation forces on every other margin.

The idea does get used, in a form the pure version does not suggest. Plane changes cost $\Delta v = 2v\sin(\Delta i/2)$, proportional to the speed at which they are made, so a large plane change is enormously cheaper far out. A bi-elliptic plane change — climb, turn, return — beats a direct plane change for inclination changes above about 39°, and that manoeuvre has been flown: it is how spacecraft are moved between very different inclinations when time is available, and it is the reason a mission to a retrograde or polar target may leave in an unexpected direction.

What the second burn is really buying #

It is worth being precise about what makes the middle burn cheap, because “a burn far out is cheap” is loose enough to be wrong in the other direction.

A burn far out is not cheap for changing the energy — it is the worst possible place for that, by the Oberth argument. What it is cheap for is changing the shape of an orbit at fixed apoapsis, and the reason is that periapsis distance is extremely sensitive to speed at apoapsis. For an orbit with apoapsis $r_a$ and periapsis $r_p$, the speed at apoapsis is

$$v_a = \sqrt{\frac{2\mu r_p}{r_a(r_a + r_p)}},$$

which goes as $\sqrt{r_p}$ for $r_p \ll r_a$. So doubling the periapsis distance requires increasing the apoapsis speed by only 41%, and 41% of a very small number is a very small number. At an apoapsis of 200 initial radii, the whole second burn of the transfer above is 0.026 of the initial circular speed — under 200 m/s for a low Earth orbit.

That is the sense in which distance is bought cheaply and energy is not. A vehicle at a great distance has almost no kinetic energy left, so what little velocity it has is entirely determining the shape of its orbit, and small absolute changes to it move the periapsis enormously.

The same leverage is what makes a distant apoapsis dangerous. A spacecraft in a highly elliptical orbit has its periapsis altitude changed by tens of kilometres by solar and lunar perturbations at apogee, which is why such orbits need station-keeping and why the Moon’s influence is a design consideration for anything reaching past a few hundred thousand kilometres.

The Oberth effect, on the other side of the argument #

The bi-elliptic works because a burn far out is cheap in the sense of achieving a large change in periapsis. The Oberth effect says the opposite thing about energy, and holding both in mind is what makes the trade legible.

The energy gained from a burn of $\Delta v$ at speed $v$ is

$$\Delta\varepsilon = v\,\Delta v + \tfrac{1}{2}\Delta v^2,$$

which is largest where $v$ is largest — deep in the gravity well. That is why an interplanetary departure burn is made at perigee, and why a mission may deliberately drop toward a planet before firing.

The two statements do not conflict, because they are about different quantities.

• Energy is bought most efficiently low down, where the vehicle is fast.
• Geometry — the direction of the velocity vector, the orientation of the orbital plane, the position of periapsis — is changed most cheaply far out, where the vehicle is slow.

A bi-elliptic transfer is a manoeuvre whose expensive part is geometric, so it moves that part outward. A departure burn is a manoeuvre whose expensive part is energetic, so it stays inward. Both are consequences of the same quadratic.

The one place a three-burn route is flown #

The pure bi-elliptic is a curiosity. Its close relative is standard practice, and the difference between them is instructive about what “optimal” is optimising.

A supersynchronous transfer orbit raises apogee beyond geostationary altitude — typically to 60,000 or 90,000 km rather than 35,786 — and then performs a combined circularisation and plane change at that higher apogee. It is a bi-elliptic in everything but name, and almost every commercial geostationary satellite launched from a high-latitude site has flown one.

The reason it pays is that the manoeuvre being moved outward is a plane change, not a periapsis raise. A launch from Cape Canaveral at 28.5° latitude leaves the vehicle in a 28.5° orbit, and geostationary orbit is equatorial, so 28.5° of inclination has to be removed. At geostationary speed of 3.07 km/s that costs

$$2 \times 3.07 \times \sin(14.25°) = 1.51\ \text{km/s}.$$

At an apogee of 90,000 km the orbital speed is under 1 km/s, and the same turn costs less than half as much. Spending a little extra to climb higher buys a large discount on the turn, and the arithmetic comes out in favour.

Baikonur, at 45.6° latitude, faces a 45.6° plane change and the case is stronger still — which is why Russian geostationary launches have used supersynchronous profiles routinely, and why launch-site latitude is a first-order commercial fact rather than a detail.

What was actually measured #

There is no observation to report here in the usual sense — this is a result about a model, not about the sky — so the honest question is what the model’s predictions have been checked against, and the answer is spacecraft navigation.

Every Δv quoted above is the difference of two speeds computed from the two-body relation. The check is whether a burn of the computed magnitude produces the predicted orbit, and the answer is that it does to within the accuracy of the propulsion system rather than of the theory.

Radiometric tracking measures a spacecraft’s line-of-sight velocity to about 0.05 mm/s by Doppler and its range to a few metres. A geostationary transfer’s apogee burn of 1,470 m/s is planned to that theory and executed to about 0.1% — and the residual is dominated by thrust-vector misalignment, propellant slosh and the finite burn duration, not by any error in the vis-viva arithmetic.

The finite burn duration is the one systematic worth naming, because it is where the idealisation genuinely fails. All the expressions above assume an impulsive burn: instantaneous, at a point. A real apogee burn from a chemical stage takes several minutes, during which the vehicle moves along its orbit and the thrust direction is no longer optimal. The resulting gravity loss is small at apogee, where the vehicle is slow and gravity is weak, and large at perigee — which is a further, practical argument for the manoeuvres this essay is about, and against departure burns from very low orbits with low-thrust stages.

Electric propulsion breaks the idealisation completely. An ion thruster’s burn lasts months, the trajectory is a slow spiral rather than a sequence of conics, and the Hohmann and bi-elliptic analysis does not apply at all — the cost of a low-thrust spiral from low Earth orbit to escape is about 7.6 km/s against the impulsive 3.2, and the whole optimisation is a different problem.

The generalisation: local optima and the shape of the cost surface #

The lesson that survives outside orbital mechanics is about optimisation rather than about orbits.

The Hohmann transfer is provably optimal within its class — two impulsive burns, coplanar, circular endpoints. Widening the class to three burns changes the answer for part of the parameter range, and widening it further changes it again: there are tri-elliptic transfers, and continuous-thrust solutions, and for some geometries a transfer that uses a third body’s gravity beats all of them by a large margin.

Every one of those is a statement of the form “the optimum over a restricted set is not the optimum”. The restriction is usually invisible in how the result is quoted — “the Hohmann transfer is the most efficient” — and it is the restriction that carries all the content.

The same shape appears in the fact that the Hohmann cost is non-monotonic. A cost function that rises and then falls means there is a worst destination, and it is worth knowing that the hardest circular orbit to reach from low Earth orbit is not the highest one: it is the one at 15.58 times the radius, about 100,000 km up. Beyond it, things get easier again. That is not intuitive and it comes straight out of the arithmetic.

The transfer that beats both, and needs a third body to do it #

The generalisation above says that widening the class changes the answer, and there is a case where widening it far enough beats every conic route by a margin large enough to have been flown in an emergency.

A spacecraft can reach the Moon by a Hohmann-like transfer in about three days, and it arrives moving fast relative to the Moon, so it needs a substantial burn to be captured. The total cost is set by that arrival burn as much as by the departure.

There is another way in. Instead of aiming at the Moon, raise apogee far beyond it — out to a million or a million and a half kilometres, where the Sun’s gravity is no longer negligible compared with the Earth’s. Out there the solar perturbation acts on the trajectory for weeks, and it can be arranged to raise the perigee and adjust the geometry so that the vehicle returns to the Moon’s distance moving slowly relative to the Moon — slowly enough to be captured with a very small burn, or in the limiting case with none at all.

That is ballistic capture, and the region of phase space it exploits is called the weak stability boundary: the fuzzy transition where a spacecraft is bound to neither body decisively and small perturbations decide which way it goes.

The saving is real. Such a route reaches lunar orbit for a few hundred metres per second less than a direct transfer, which for a small spacecraft is a substantial fraction of its propellant. The price is time: three to five months rather than three days.

It was flown, and it was flown as a rescue. A Japanese spacecraft launched in 1990 had failed to release its lunar orbiter, and was left without the propellant for a conventional capture. A low-energy route was designed for it, and it reached lunar orbit in 1991 on what it had left.

The manoeuvre exists only because the two-body model is wrong, which makes it the exact complement of everything else in this essay. Every threshold above was computed in a model with one gravitating body; the route that beats them all is invisible in that model, because it is made entirely out of the perturbation the model discards.

Where the model stops #

Impulsive burns. Every expression assumes instantaneous velocity changes. Real burns take time, and the correction is not negligible in low orbit.

Coplanar and circular. The endpoints are assumed circular and in one plane. Real transfers combine a plane change with the transfer burns, and the optimal split between them is a separate calculation that changes both thresholds.

Two bodies. Above a few hundred thousand kilometres from the Earth, the Sun’s and Moon’s gravity are not negligible, so a bi-elliptic transfer with an apoapsis at a million kilometres is not on an ellipse at all, and three bodies have no closed solution. That is a genuine problem for exactly the regime where the bi-elliptic wins.

The figures draw both routes as complete curves, and the flight times are nowhere on them. The orbits figure shows the bi-elliptic path as a larger drawing, which understates the trade badly: the two routes differ by a factor of fifteen in duration and by under one per cent in cost, and only one of those is visible.

The ladder from here #

Later rungs on this anchor: the Hohmann transfer’s optimality proof, and the class it is optimal within. The bi-elliptic thresholds derived rather than found numerically. Plane changes, and the 38.94° crossover. Combined transfer-and-plane-change optimisation. Low-thrust spirals and the Edelbaum equation. Weak-stability-boundary transfers, which use the Sun’s perturbation to arrive at the Moon for less Δv than any conic route. Gravity losses and finite-burn corrections.

The bi-elliptic was described by Ary Sternfeld in 1934, a year after Hohmann’s book had established the two-burn result — so the exception arrived almost immediately, and has been a curiosity for ninety years. It is the standing example of a manoeuvre that is optimal, publishable, correct, and has never once been worth flying.