One trajectory, stitched from three two-body problems

A spacecraft going from Earth to Mars is under the gravity of the Sun, the Earth and Mars simultaneously, for the entire journey. That is a four-body problem, and it has no closed-form solution — not because nobody has found one, but because none exists.

Yet interplanetary trajectories were designed on slide rules. Every Mariner, Pioneer and Voyager mission was planned with a method that gives answers good to a fraction of a per cent, using nothing but the exactly solvable two-body problem.

The method is to lie, in a controlled way. Pretend that at any moment exactly one body is pulling. The trajectory then becomes a sequence of conics — a hyperbola about the Earth, an ellipse about the Sun, a hyperbola about Mars — each of them exact, joined at boundaries where the pretence is switched. Those boundaries are the spheres of influence, and the method is patched conics.

Where the boundary comes from #

The sphere of influence is not a place where one gravity exceeds another. That surface exists — it is where the Sun’s and the Earth’s attractions are equal, about 259,000 km from the Earth, at a radius the shell theorem marks nothing special about — and it is the wrong boundary, which is worth understanding because the wrong answer is the intuitive one.

The right question is not “which pull is larger” but “which two-body problem is less perturbed”. Compare two descriptions:

• Planet-centred. The spacecraft orbits the planet, and the Sun is a perturbation. The perturbing acceleration is the tidal difference between the Sun’s pull on the spacecraft and on the planet — not the Sun’s full pull, because a common acceleration does not disturb a relative orbit.
• Sun-centred. The spacecraft orbits the Sun, and the planet is a perturbation, at its full strength.

Setting the two perturbation ratios equal gives Laplace’s result:

$$r_{\text{SOI}} = a\left(\frac{m}{M}\right)^{2/5}.$$

The exponent is $2/5$, not the $1/2$ that equating forces would give, and it is the tidal cancellation that produces it. For the Earth the answer is 924,000 km — three and a half times further out than the equal-force surface, and about 2.4 times the distance to the Moon.

The boundary is a mathematical device rather than a physical surface. Nothing happens there. It is the radius at which it becomes more accurate to change which fiction is being used.

The three legs #

The departure hyperbola. The spacecraft leaves a parking orbit with a burn that puts it on a hyperbola about the Earth. What matters at the far end is the residual speed, $v_\infty$, and the quantity budgeted is its square, the characteristic energy $C_3 = v_\infty^2$. A minimum-energy Mars transfer needs $C_3 \approx 9$ km²/s²; a trans-lunar injection needs about $-2$, negative because it stays bound.

The heliocentric ellipse. At the edge of the sphere of influence the spacecraft is handed to the Sun. Its heliocentric velocity is the Earth’s orbital velocity plus $\mathbf{v}_\infty$ as a vector — and the vector addition is the entire trick, because a departure in the direction of the Earth’s motion adds 29.8 km/s of free velocity that no propulsion supplied.

The arrival hyperbola. At Mars’s sphere of influence the spacecraft is handed to Mars, again by subtracting the planet’s heliocentric velocity from its own. The result is a hyperbola whose $v_\infty$ is fixed by the transfer, and whose periapsis is chosen by aiming, which is what turns a flyby into a capture or a slingshot.

The vector addition that makes the whole thing work #

The middle step deserves more than a sentence, because it is where the free velocity comes from and it is the step most often drawn wrongly.

At the edge of the Earth’s sphere of influence, the spacecraft has a velocity $\mathbf{v}_\infty$ relative to the Earth, of a few kilometres per second. The Earth has a velocity of 29.78 km/s relative to the Sun. The spacecraft’s heliocentric velocity is the vector sum of the two:

$$\mathbf{v}_{\text{helio}} = \mathbf{v}_{\oplus} + \mathbf{v}_\infty.$$

Because the sum is a vector sum, direction is worth as much as magnitude. A departure aligned with the Earth’s motion produces a heliocentric speed of $29.78 + v_\infty$ — the whole of the Earth’s orbital speed, free. A departure at right angles produces $\sqrt{29.78^2 + v_\infty^2}$, which for $v_\infty = 3$ km/s is 29.93 — a gain of 0.15 km/s instead of 3.

So the launch geometry is not a detail. The direction in which a spacecraft leaves the Earth’s sphere of influence determines the heliocentric orbit almost entirely, and that direction is set by when in the year the launch occurs and by the hyperbola’s asymptote. That is what a launch window is: the set of dates on which a departure of achievable $C_3$ points the asymptote in a direction that produces the wanted heliocentric ellipse. A window closes not because a planet has moved out of reach but because reaching it would require a $C_3$ the launch vehicle cannot deliver. For Mars the pattern repeats every 25.6 months, which is the synodic period of the two orbits, and every window is a little different because both orbits are elliptical and their relative geometry does not repeat exactly.

Why the errors are small #

The method makes two errors at each patch, and both are small for the same reason.

The position error. The sphere of influence is treated as a point, so the spacecraft is assumed to be at the planet’s position when it is handed over. For the Earth that is an error of 924,000 km on a journey of 400 million — two parts in a thousand.

The velocity error. The velocity is patched by adding vectors, ignoring the fact that at the boundary the spacecraft has already been decelerated by the planet’s gravity on the way out. That error is of order $v_{\text{esc}}(r_{\text{SOI}})$, which for the Earth is 0.93 km/s against a heliocentric speed of 32 km/s — three per cent, and it partly cancels between departure and arrival.

Both errors shrink as the target gets further away, which is the reason the method was good enough for the missions that established it.

Set against that, the errors are worst for the shortest journeys — a transfer to Venus, a lunar trajectory, a rendezvous within the Earth–Moon system — and it is precisely there that the two-body approximation is usually abandoned in favour of a restricted three-body treatment. The method is not uniformly good; it is good in proportion to how much of the path is dominated by one body, and the ratio of the sphere of influence to the transfer length is the number that says so.

Both errors are systematic and both can be corrected. The standard practice is to use patched conics to find the trajectory — the launch date, the transfer geometry, the approximate Δv — and then to hand the result to a numerical integrator that models every body at once, which refines it into the trajectory actually flown. The conic solution is the initial guess that makes the numerical problem converge; without it the search space is intractable.

That division of labour has not changed since the 1960s. Patched conics do the design; numerical integration does the navigation.

What the patch throws away, and when it matters #

It is worth being concrete about the discarded terms, because their size decides everything about when the method can be trusted.

Inside the Earth’s sphere of influence, the Sun’s neglected influence is a tidal term of order

$$\frac{2GM_\odot r}{a_\oplus^3},$$

which at the boundary is about $1.5\times 10^{-5}$ m/s² — small against the Earth’s own $2.9\times 10^{-4}$ m/s² there, but not negligible over the days a spacecraft spends crossing. Integrated, it displaces the outbound asymptote by a fraction of a degree.

Outside, the Earth’s neglected influence falls as the inverse square and is gone within a few million kilometres.

Both errors are largest at the boundary itself, by construction: the boundary is placed exactly where the two neglected terms are equally bad. That is a slightly uncomfortable property — the model is worst precisely at the seam — and it is the reason the position and velocity discontinuities at the patch are the standard diagnostic for whether a conic solution needs refining before it is handed to an integrator.

There is a case where the discarded term is not small and is not an error, and it is the one worth remembering. A gravity assist works because the spacecraft’s heliocentric energy changes while its planet-relative energy does not. In the patched-conic picture that is a discontinuity in one frame and a continuity in the other, and it is entirely real. The manoeuvre is not an artefact of the decomposition; the decomposition is what makes it computable.

What was actually measured #

The best evidence for the method is a mission that was designed with it and then executed for twelve years without a correction the method did not anticipate.

Voyager 2 used four gravity assists — Jupiter, Saturn, Uranus, Neptune — over a route that was found by patched-conic search across thousands of candidate launch dates and flyby geometries. The Grand Tour alignment recurs every 175 years, and finding it at all required exactly the kind of cheap approximate calculation that a conic decomposition makes possible.

The delivery accuracy is the number that tests the machinery. At Neptune, twelve years and 4.4 billion kilometres after launch, Voyager 2 passed within about 100 km of its aim point and within a few seconds of its predicted time. Most of that precision came from the numerical navigation and from twenty-odd trajectory correction manoeuvres totalling well under 100 m/s — but the trajectory being corrected to was the conic design, and the corrections were small because the design was nearly right.

The measurement that most directly checks the sphere-of-influence idea is radiometric tracking through a flyby. During Voyager 2’s Neptune encounter the Doppler residuals were used to measure Neptune’s $GM$ to about one part in $10^4$, and Triton’s to a few per cent — a mass measurement obtained by fitting a hyperbola to the observed velocity change. That fit works because the encounter genuinely is a two-body problem to within the tracking precision, for the few days it lasts.

The limits show up where the assumption fails. In the Earth–Moon system the mass ratio is 81, not $10^6$, so the Moon’s sphere of influence is 66,000 km — a sixth of its distance from the Earth — and neither body is negligible over much of the journey. Patched conics give lunar trajectories good to a few per cent, which is enough for a first design and not enough to fly. The Apollo missions were flown on numerically integrated trajectories throughout.

The hyperbola is where all the choices are #

The two hyperbolic legs occupy days out of a journey of months, and almost every decision a mission makes is taken in them.

A hyperbola about a planet is fixed by two numbers: its $v_\infty$, which the heliocentric transfer hands it and which cannot be changed without propellant, and its periapsis distance, which is chosen by aiming and costs nothing. Every consequence follows from the second.

$$v_p = \sqrt{v_\infty^2 + \frac{2\mu}{r_p}}, \qquad \sin\frac{\delta}{2} = \frac{1}{1 + r_p v_\infty^2/\mu},$$

with $\delta$ the turn angle. Aim closer and the flyby turns the spacecraft further; aim further out and it barely bends. A capture burn’s cost is $v_p - v_{\text{orbit}}$, so aiming closer also makes capture cheaper — by the Oberth effect, which is the same statement.

So the arrival hyperbola is a one-parameter family, and the parameter is free. That is why the aim point is specified so precisely — in the B-plane, a plane through the planet perpendicular to the incoming asymptote, with the target given as a two-dimensional offset — and why trajectory correction manoeuvres of a few centimetres per second, made weeks out, are what actually deliver a mission.

Patches inside patches #

The method has been described with one boundary at each end, and a real mission at a giant planet has several nested inside each other.

A spacecraft touring Jupiter’s moons is inside the Sun’s domain, inside Jupiter’s sphere of influence, and — briefly, during each encounter — inside a moon’s. The moons’ spheres are small: Ganymede’s is about 24,000 kilometres, against its own radius of 2,600 and its distance from Jupiter of a million. So a flyby of Ganymede is a hyperbola about Ganymede lasting a few hours, embedded in an orbit about Jupiter lasting weeks, embedded in an orbit about the Sun lasting years.

The hierarchy is what makes a tour computable. Each encounter is a patch of the same kind as a planetary flyby, with the planet playing the part the Sun plays in an interplanetary transfer, and the tour design proceeds by choosing a sequence of encounters and solving each leg as a two-body problem about Jupiter.

What is different is the mass ratio. Ganymede’s mass is a ten-thousandth of Jupiter’s, which is a hundred times larger a ratio than a planet’s to the Sun’s, so the moon’s sphere of influence is proportionately larger relative to its orbit and the patch is correspondingly cruder. A Jovian tour designed by patched conics needs numerical refinement sooner and more heavily than an interplanetary transfer does.

The decomposition is recursive and its accuracy degrades at every level, which is the reason a tour is designed as a sequence of conic legs and flown as a continuously integrated trajectory with a correction manoeuvre after every encounter.

The generalisation: solving a problem by cutting it up #

Patched conics is an instance of a strategy that appears wherever an intractable problem contains regions in which a tractable approximation is good.

The strategy has three parts, and they are visible here in an unusually clean form:

1. Identify the regimes in which different simplifications hold.
2. Solve each exactly, in its own natural variables.
3. Match at the boundaries, accepting an error that is small because the boundaries were chosen to make it small.

Matched asymptotic expansions in fluid dynamics do exactly this: the boundary layer near a surface and the inviscid flow away from it are solved separately and matched in an overlap region. The Born–Oppenheimer approximation separates nuclear and electronic motion in a molecule because their timescales differ by the square root of a mass ratio, and matches at the end. The WKB approximation treats a wave as locally plane wherever the potential varies slowly and patches across the turning points where it does not.

In every case, the small parameter that justifies the decomposition is what sets both the size of the error and the location of the boundary — and in every case there is a region where the method is known to be wrong and is used anyway, because the error there is bounded and correctable.

There is a fourth part that is easy to leave implicit and is worth stating: the decomposition has to be checked against the undivided problem somewhere, or the error estimate is a hope. For patched conics that check is numerical integration, and it is run for every mission.

One more thing the patch buys is worth naming, because it is not accuracy. A patched-conic trajectory is analytic: the departure hyperbola, the heliocentric ellipse and the arrival hyperbola each have closed-form solutions, so a change to the launch date propagates through the whole design in microseconds rather than in an integration. That is what makes a porkchop plot possible — hundreds of thousands of complete trajectories evaluated across a grid of two dates — and no numerical integrator has ever been fast enough to build one directly. The design space is searched in conics and the answer is refined in integrations, and the reason is not that conics are more accurate but that they are four or five orders of magnitude cheaper. The same argument explains why the method survived the arrival of computers that could integrate anything: the cost that mattered was never the cost of one trajectory but the cost of the millionth one, and that cost is what decides whether a mission’s design space can be searched at all rather than merely sampled, which is the difference between finding the best launch window and finding a launch window that works.

Where the model stops #

The Earth–Moon system. At a mass ratio of 81 the decomposition is marginal, and lunar missions have never been flown on it.

Low-thrust trajectories. A continuously thrusting vehicle is not on a conic anywhere, and the whole method is inapplicable. Dawn’s spiral to Vesta and Ceres was optimised by entirely different techniques.

Weak-stability-boundary transfers. There is a class of lunar trajectory that reaches the Moon for less Δv than any conic route by exploiting the Sun’s perturbation near the edge of the Earth’s sphere of influence — the region where patched conics is least accurate. Hiten in 1991 and GRAIL in 2011 flew these. The method that is wrong there is precisely the method that cannot find them.

Resonant and chaotic regions. Near a libration point or in a mean-motion resonance, no single body dominates and the trajectory can be sensitive to initial conditions in the way three bodies generally are.

The figures cannot draw the spheres of influence honestly and legibly at once. The trajectory figure draws them at true relative scale, which makes them nearly invisible — correct, and it makes the point that the patches are small. Any drawing that makes them large enough to inspect has necessarily exaggerated them by a factor of ten or more, and every textbook diagram of this method does exactly that.

The ladder from here #

Later rungs on this anchor: the sphere of influence derived from the two perturbation ratios in full. $C_3$, $v_\infty$, and launch-vehicle performance curves. The B-plane, and how an arrival is aimed. Porkchop plots, and what a launch window is. Lambert’s problem, which is what actually gets solved to find the transfer. Gravity assists as vector additions across a patch. Weak-stability-boundary transfers and the invariant manifolds that organise them. Low-thrust trajectory optimisation.

The seams are the interesting part of this method, and the reason is a general one. The place where a model is known to be false is the place where the next model is found — weak-stability-boundary transfers exist in exactly the region patched conics handles worst, and they were found by people looking at what the approximation was throwing away.