The discontinuity a patched conic hides

An interplanetary trajectory is a solution of an n-body problem, and nobody designs one that way. What is designed is a chain of two-body problems: a hyperbola away from the launch planet, an ellipse around the Sun, a hyperbola towards the target. Each piece is exact, each is solved in closed form, and the three are joined at a surface called the sphere of influence.

The joins are where the method is not true. At the boundary the term being neglected on one side is at its largest, and so is the term being neglected on the other.

One trajectory stitched from three two-body problems is the method; this essay is about the stitches, and about why an approximation with a large, known, computable error is the right tool for the job it is doing.

Where the surface comes from #

The sphere of influence is not a physical boundary and it is not the Hill sphere. It answers a specific question: at what distance from the planet is it equally bad to treat the planet as a perturbation on a heliocentric orbit and to treat the Sun as a perturbation on a planetocentric one?

Both errors are ratios of a perturbing acceleration to a primary one. Writing them out and setting them equal gives Laplace’s result,

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

with the two-fifths power coming from the different ways the two perturbations scale — one as the inverse square of the separation and one as the separation itself, because a tide is a difference of accelerations.

That exponent is worth pausing on. The Hill sphere, which asks a different question — where can a moon remain bound — has an exponent of one third. The two radii differ by a factor of a few and they answer genuinely different questions: one is about what stays, the other about which approximation is less wrong.

There is a third radius in the same family that is worth naming to keep the three apart. The Hill sphere asks where a small body can remain bound to the planet against the Sun’s tide, and its exponent is one third. The sphere of influence asks which of two approximations is less wrong, and its exponent is two fifths. The Laplace radius is another name for the second. All three are of the same order for any given planet and they differ by factors of two or three, and using one where another is meant is a common source of confusion in the literature — particularly because the Hill sphere is the larger of the two for every planet in the solar system.

Why the error peaks at the join #

At the sphere of influence the neglected solar tide is a definite fraction of the planet’s attraction, and the fraction turns out to be exactly twice the fifth root of the mass ratio — about 0.09 for Mercury and 0.50 for Jupiter.

So the surface does not equalise the two accelerations across the planets; it equalises the ratio of the two perturbations, which is a different statement. Inside the surface the planet dominates and the error made by ignoring the Sun grows outward; outside, the reverse. The two errors cross at the surface, and both are at their maximum for their respective regimes exactly there.

The consequence is that a patched conic is least accurate at the one place where the two pieces have to agree. That is a structurally awkward property and it is not an accident of the definition: any surface separating the two regimes has it, because the two errors are monotonic in opposite directions.

One more property of the error is worth having, because it decides where in a mission it matters. The velocity error accumulated inside a sphere of influence is not a random walk — it is a systematic bias with a definite direction, set by the geometry of the encounter, and it therefore does not average away over multiple encounters. A trajectory with three flybys accumulates three such biases, and because each flyby amplifies whatever error it arrives with, the third one arrives with an error much larger than the sum. The amplification factor at a close flyby is of order the ratio of the approach speed to the change in it, and it is routinely a factor of ten or more per encounter.

The approximation inside the approximation #

There is a further simplification that almost every broad trajectory search makes, and it is worth knowing because it shrinks the discontinuity to a point rather than removing it.

In a linked-conic search the sphere of influence is taken to have zero radius. The flyby then happens instantaneously and at the planet’s own position: the spacecraft arrives with a hyperbolic excess velocity, the encounter rotates that vector through a turn angle set by the closest approach, and the spacecraft leaves with the same speed relative to the planet in a different direction. The heliocentric velocity changes because the planet’s own velocity is added back on either side, and the whole encounter costs no time and no distance.

That is a cruder approximation than the patched conic and it is the one used for the searches that matter, because it makes a flyby a single algebraic operation on a vector. A search over launch dates and encounter dates for a four-planet sequence is then a search over a handful of numbers with a closed-form evaluation, which is what makes an exhaustive sweep of a decade of launch opportunities possible at all.

The error it introduces is bounded and computable in the same way as before. The encounter takes a real time — a day or two at Jupiter, hours at Venus — during which the planet moves along its own orbit and the spacecraft moves with respect to the Sun. Collapsing that to an instant misplaces the encounter by the distance the planet travels in that time, which at Jupiter is a few hundred thousand kilometres. Against a sphere of influence of forty-eight million kilometres that is a small displacement, and against a targeted flyby altitude of a few thousand kilometres it is enormous — so the linked conic finds the sequence and says nothing usable about the aim point.

The hierarchy is therefore three deep rather than two, and each level exists because the level above it cannot afford the level below. A linked conic searches sequences; a patched conic refines one sequence into a trajectory with real encounter geometry; an n-body integration turns that into something that can be flown. Each hands the next a starting guess inside its basin of convergence, and the whole chain is designed around what each level’s error is rather than around making any level accurate.

There is one place where the zero-radius approximation does something the fuller one cannot, and it is worth recording because it looks like a defect. Because the flyby is instantaneous, the linked conic makes the change in heliocentric speed a pure function of the turn angle, which in turn is a pure function of the closest-approach distance and the excess speed. That is exactly the relation a designer wants to reason about — how much speed can this planet give, at this arrival velocity, without flying below its atmosphere — and it is visible in the algebra rather than buried in an integration. The approximation that discards the most is the one that exposes the trade.

What the method is nevertheless for #

None of this makes patched conics wrong to use, and it is worth being clear about what it is used for.

A trajectory design is a search over a large space: launch date, arrival date, flyby sequence, manoeuvre placement. Evaluating a candidate has to be cheap, because millions are evaluated. A patched-conic evaluation is a few closed-form solves and takes microseconds; an n-body integration takes seconds. The search is only possible with the fast evaluation.

What the search produces is a neighbourhood: a launch window, an approximate arrival geometry, an approximate manoeuvre schedule. The real trajectory is then found by differential correction — integrating the full n-body equations, comparing against the desired end conditions, and adjusting the initial state by a Newton step until it converges. The patched conic supplies the starting guess, and the guess is close enough that the correction converges in a few iterations.

That division of labour is the whole architecture of mission design, and it survives precisely because the patched conic’s error is metres a second rather than kilometres.

There is a second use, and it is the one that makes the method indispensable rather than merely convenient: it supplies the vocabulary. A patched conic decomposes a trajectory into quantities with names and meanings — a hyperbolic excess velocity, an arrival asymptote, a B-plane target, a turn angle at a flyby. Those are the quantities a mission is specified in, negotiated in and flown by, and they exist only because the two-body problems have closed-form solutions with named parameters. An n-body integration produces a state vector and no vocabulary at all. Aiming at a plane rather than at a planet is a piece of that vocabulary, and it is a patched-conic construction used to describe an integrated trajectory.

What was actually measured #

The error is not measured in the sky; it is measured by doing the calculation both ways, and the comparison is routine enough to be a standard verification step.

Integrate and compare. Take a patched-conic trajectory, integrate the same initial state under the full gravitational model, and measure the divergence at the target. For a typical Earth–Mars transfer the arrival position differs by tens of thousands of kilometres — which sounds catastrophic and is exactly the size of the correction the differential corrector removes in its first iteration.

Vary the surface. Repeating the patch at a boundary twice or half the Laplace radius changes the answer, and the change is a direct measurement of the method’s error. The answer is least sensitive near the Laplace radius, which is the practical justification for that definition rather than a philosophical one.

And the navigation confirms it in flight. The trajectory actually flown is the integrated one, and the tracking residuals against it are metres. Against the patched-conic design they would be thousands of kilometres. The difference between those two numbers is what this essay is about.

And the discontinuity itself is visible in the design. A patched-conic trajectory has a velocity that is continuous at the boundary by construction and an acceleration that is not — the planet’s pull switches on and the Sun’s switches off at a surface. Plotting the acceleration along the designed trajectory shows a step of exactly the size the previous figures give, and the step is the error. It is one of the few cases where an approximation’s error is directly visible in its own output rather than requiring a comparison against a better calculation.

Where the picture stops #

The picture stops in three places, and the second is where the method breaks entirely.

It ignores every other body. The patch handles the Sun and one planet; a real trajectory feels the other planets, the Moon, the largest asteroids, solar radiation pressure and the planet’s own oblateness. All of those are smaller than the neglected tide over most of a transfer and none is smaller near a target.

Low-energy trajectories are outside it. A transfer that arrives at the sphere of influence with almost no excess speed spends an unbounded time near the boundary, and the accumulated error diverges. Those trajectories — ballistic capture, weak stability boundary transfers, the ones that exploit the tubes threading a Lagrange point — cannot be designed with patched conics at all, and their discovery required a completely different set of tools.

And the sphere is a sphere. The region in which a planet dominates is not spherical: it is elongated along the Sun–planet line, because the tide is compressive in one direction and stretching in another. Using a sphere is a convenience, and the departure from sphericity is a further error of the same order as the one already being made.

One more limit deserves stating because it is the one that decides whether the whole architecture works. The differential correction that turns a patched-conic guess into a real trajectory is a Newton iteration, and Newton iterations converge only from within a basin. For ordinary transfers the basin is comfortably larger than the patched-conic error and the correction converges in three or four steps. For a trajectory with several flybys the sensitivity compounds — a metre a second at the first encounter is thousands of kilometres at the third — and the basin shrinks accordingly. Multi-flyby trajectories are therefore corrected in stages, one encounter at a time, and the design process is as much about managing the convergence as about finding the path. Stealing speed from a planet is easy to describe and hard to fly for exactly this reason.

Why an approximation with a known error is better than a better approximation #

The general lesson is worth separating out, because the instinct to improve the approximation is usually wrong.

An approximation used to generate a starting guess needs to be fast and roughly right, not accurate. Improving a patched conic — by adding a tidal term, by using a non-spherical boundary, by iterating the patch — costs computation and buys accuracy that the differential corrector was going to supply anyway. The corrector converges from a bad guess about as fast as from a slightly better one, because Newton’s method is quadratic once it is close.

What the improvement would buy is a larger basin of convergence, and that does matter for the hard cases: a low-energy trajectory whose patched-conic guess is so wrong that the corrector diverges is a real problem, and it is the reason those trajectories were not found until the tools changed.

A second reason not to improve it is that the improved version stops being a two-body problem, and the closed form is the whole value. Add a tidal term and the hyperbola is no longer a hyperbola, the arrival asymptote is no longer defined, and the vocabulary in the previous section evaporates along with the algebra.

The architecture is worth recognising as a pattern rather than a mission-design detail. A fast approximate method to search a space, an expensive exact method to refine a candidate, and a correction step whose convergence depends on the approximation being in the right basin: that is how nearly every hard optimisation in physical science is done. The same relationship holds between a perturbation series and a numerical integration — the series says what the structure is and the integration says what the answer is, and neither replaces the other.

One more observation about the size of the numbers, because it is easy to lose track of what is large. The patched-conic velocity error is metres a second. A mid-course correction manoeuvre on a typical interplanetary mission is tens of metres a second. The navigation’s own uncertainty before a correction is centimetres a second. So the approximation’s error is small compared with what the mission was going to spend anyway and enormous compared with what the tracking can see — which is exactly the regime in which an approximation is useful for design and useless for operations, and the reason the two activities use different tools on the same trajectory. The same division exists between an analytic perturbation theory and a numerical integration, and for the same reason.

Where the ladder goes next #

The natural next rung is the differential correction itself: how a trajectory is adjusted onto a target by Newton iteration on a state transition matrix, and why the sensitivity of the arrival to the departure — which is enormous — is the quantity that decides how well a mission can be flown. The rung after it is what happens when the patched conic has no valid starting guess to offer, which is the regime of the low-energy transfers and of the manifolds that thread the necks between regions.