A trajectory with no good first guess

A spacecraft on its way to Mars arrives seventy-six kilometres from where it was aimed for every millimetre a second its departure was wrong. That is a large number, and it is also a tame one. It is the same at every point along the way to within a factor of a few, it grows only in proportion to the flight time, and it is the derivative that Newton’s method divides by when it corrects a trajectory onto its target. Because it is tame, a two-body sketch — three conics stitched together — is a good enough first guess, and two iterations turn the sketch into a real trajectory.

There is a class of trajectories for which none of that is true. The cheapest routes between the Earth and the Moon, the transfers that let a spacecraft drift into lunar orbit for almost nothing, and the paths that move between the Sun–Earth Lagrange points all spend days to weeks near the necks of the three-body problem, where the forbidden region opens at a Lagrange point and a trajectory may pass. Near a neck the derivative Newton’s method needs is not tame. It grows exponentially, and everything that depended on it being polynomial — the quality of the first guess, the digits of the computation, the convergence of the correction — fails in a characteristic way.

The matrix Newton divides by #

A correction scheme asks one question: if the departure changes by a little, how does the arrival change? The answer is a matrix, the state-transition matrix, whose entries are the partial derivatives of the final position and velocity with respect to the initial ones. It is obtained not by trying many departures but by integrating a second set of equations alongside the trajectory — the variational equations, which say how a small displacement from the trajectory evolves — and a correction step then solves a linear system with it: the change in departure that cancels the observed miss is the miss divided by the matrix.

On a two-body arc the matrix grows slowly. A small error in speed is an error in orbital period, and a period error makes the spacecraft fall behind or run ahead of where it should be by an amount that grows in proportion to the time. That secular growth, plus oscillations at the orbital period, is all a conic does to an error, and in the figure the green curve rises from a tenth to about ten over two weeks.

Near a collinear Lagrange point the linearised dynamics have a pair of real eigenvalues, $\pm\lambda$, and the matrix’s largest entries grow as $e^{\lambda t}$. In the Earth–Moon system $\lambda$ is 2.93 in the system’s natural units, which is an e-folding every 1.48 days; integrated along the point itself, the variational equations reproduce that rate to within two per cent, and over two weeks the amplification is more than two thousand. The difference between the curves is not a matter of degree. A polynomial and an exponential that agree for the first four days disagree by a factor of two hundred and fifty after fourteen.

The physical reason is the one that makes the region interesting. A collinear point is a saddle of the effective potential: one direction leads downhill towards the Earth, the opposite one downhill towards the Moon, and a spacecraft balanced near the top runs off one way or the other at a rate set by the curvature of the saddle. Errors in where it was put are amplified at that rate, which is exactly the rate at which an uncorrected spacecraft at the point drifts away. For holding station that rate sets how often a burn is needed. For designing a trajectory it sets something worse.

The digits run out #

The first casualty is precision.

Double-precision arithmetic carries about sixteen significant digits. Every e-folding of the amplification uses up 0.43 of them — the logarithm of e to base ten — because an error in the sixteenth digit of the departure has become an error in the fifteenth digit of the position. Near the Earth–Moon L1 point that happens every day and a half, and after fifty-four days nothing of the departure survives: the computed position is determined by rounding error. No integrator, however accurate per step, can compute such a trajectory forward from its start as a single piece. On the conic the same arithmetic loses only the few digits that the secular growth costs.

A low-energy transfer to the Moon spends weeks to months in the region where the Earth’s and the Sun’s or the Moon’s pulls are comparable. It is therefore a trajectory whose end is not a computable function of its beginning in double precision. That sounds like a paradox about trajectories that spacecraft have flown, and it is not: they were flown with corrections along the way, which reset the amplification each time, and they were designed by a method that never asked the beginning to determine the end.

How good the first guess must be #

The second casualty is the correction itself.

Newton’s method converges when the first guess is close enough that the linear model of the trajectory — the state-transition matrix — is still roughly right over the size of the correction. Near a neck the model is right only over a range that shrinks as the amplification grows, because a correction that is a modest change in departure becomes a large change in path where the dynamics are nonlinear. Each point in the figure is a full iteration on the integrated three-body motion, started from guesses of increasing error until it stops returning to the right answer. A trajectory that lingers a day near the neck can be found from a guess wrong by a kilometre a second; one that lingers a week needs its guess within eighteen metres a second. For a month of lingering the required accuracy is beyond anything a sketch could supply.

A patched conic is a sketch. It ignores the third body exactly where the third body matters most, and the error it makes there is not metres a second but the whole difference between a trajectory that passes through the neck and one that turns back. For a Mars transfer the correction converged in two iterations from a two-body guess; near a neck it does not converge at all, because the guess is outside a basin that has shrunk faster than any improvement to the sketch could follow.

Cutting the trajectory into pieces #

The cure has been known for as long as the problem, and it is a change of question rather than a better guess.

Instead of asking which departure produces the right arrival, ask for a sequence of states along the way such that each, propagated over a short interval, lands on the next. Each short interval amplifies errors only by $e^{\lambda T/N}$ for $N$ pieces, so each link of the chain is as well conditioned as a conic arc, and the whole system of equations — more unknowns, since every join is now an unknown state — is one Newton’s method can solve. The figure measures it directly: the same twelve days, cut into twelve one-day pieces, gives each piece a basin more than a hundred times wider. That is multiple shooting, and every trajectory through a Lagrange-point region is computed this way: by a chain of short segments stitched together, with the continuity at the joins enforced as constraints.

The stitching is the same idea as the patched conic’s, turned round. A patched conic stitches exact solutions of an approximate problem; multiple shooting stitches approximate starts of the exact problem. In both the seams are where the work is. And it moves the difficulty rather than removing it: the question is no longer how good the first guess must be but where the joins should go, and a chain of segments started from nonsense converges to nonsense.

Walking out along a family #

There is a second way round the shrinking basin, and it is the one used to find the periodic orbits near the Lagrange points in the first place. Instead of guessing the hard trajectory, solve an easy one and change it gradually. A periodic orbit of very small amplitude around a Lagrange point is given almost exactly by the linearised dynamics, so the first member of the family needs no guess at all; increase the amplitude a little, use the previous orbit as the first guess, and Newton’s method converges because the change is small; repeat. The family is walked out step by step to the large orbits that missions actually use, each step inside the basin of the next.

The same continuation works along any parameter. A transfer can be found first for a mass ratio where the necks are wide and the lingering is short, and the mass ratio lowered step by step to the real one; or for a short time of flight, lengthened day by day. The basin figure above says how small each step must be: the basin shrinks by roughly a factor of e for each e-folding time added, so a step that adds a fraction of an e-folding time stays inside it. Continuation is slow, and it finds only the solutions connected to the easy one it started from, but it never needs a guess better than the last answer.

Where a guess can come from #

So the guess for the chain must come from somewhere that already knows the neck. It comes from the structure of the three-body problem itself: the periodic orbits around the Lagrange points and the tubes of trajectories that leave and approach them, which are computed from the same linearisation that makes the neck unstable. A trajectory built by joining a tube leaving the Earth’s region to a tube arriving in the Moon’s has the right qualitative path from the start — it passes through the neck because the tubes do — and a multiple-shooting correction then turns it into a real trajectory, as the patched conic turns into one for a Mars transfer. The instability that makes the problem ill-conditioned is the same instability whose geometry supplies the guess.

That is the general shape of what happens in a chaotic region: the sensitivity that makes forward prediction hopeless is a statement about directions, and the directions are computable. A trajectory designer working near a neck uses the unstable directions as a map rather than fighting them as noise.

Why the cheap trajectories are the hard ones #

It is not an accident that the trajectories with no good first guess are the cheap ones. A transfer saves propellant by arriving with as little excess energy as possible, and in the three-body problem the least energy that will carry a spacecraft from one body’s region to the other’s is the energy at which the forbidden region’s neck just opens. A trajectory with barely enough energy to pass the neck passes it slowly, close to the saddle, and spends a long time where the saddle’s instability acts on it. The saving and the ill-conditioning are the same fact: the less energy the spacecraft has to spare, the longer it lingers where errors grow exponentially.

The missions that have used these routes show the trade in their flight times. The Japanese probe Hiten reached lunar orbit in 1991 on a transfer that went out to four times the Moon’s distance and fell back, saving enough propellant to make a mission possible that a direct transfer could not have flown. GRAIL’s two spacecraft took three and a half months to reach the Moon in 2011 by way of the Sun–Earth L1 point, where a direct transfer takes three days, and arrived needing much less propellant to be captured. CAPSTONE took four months in 2022 to reach a lunar halo orbit, on a ballistic transfer that loops out a million and a half kilometres from the Earth. Each of those months near a neck is many e-foldings of the amplification drawn here, and each trajectory was designed in pieces and flown with corrections.

The spacecraft that take the direct routes are paying for conditioning as much as for time. A three-day transfer to the Moon is a nearly two-body arc, its correction converges from a conic in a couple of iterations, and its errors grow tamely between burns. The cheap route gives up all of that, and its cost appears not on the propellant budget but on the design and navigation teams’ calendars.

The same matrix in flight #

The state-transition matrix does not stop mattering once the trajectory is designed. In flight the spacecraft’s state is estimated from tracking — ranges and Doppler shifts measured from the ground — and the estimate’s uncertainty is carried forward between measurements by the same matrix. Near a neck the uncertainty grows exponentially between tracking passes, and it grows fastest in one direction, the unstable one, so that the error ellipsoid becomes a long needle pointing along the direction the spacecraft would run off in — the same stretching that makes an orbit determined from short arcs uncertain mostly in one direction, but far stronger.

That sets the navigation cadence for these missions. Tracking has to be frequent enough that the uncertainty is measured before it has grown by more than a few e-foldings, and corrections have to be applied before the needle has turned into a choice between two outcomes — passing through the neck or turning back. A spacecraft near the Earth–Moon neck can lose its trajectory in a week without tracking; one near the Sun–Earth points, in months. Both numbers are the e-folding time multiplied by the number of e-foldings the mission can afford.

The same exponential, on a slower clock #

The Earth–Moon numbers are not special. The instability near a collinear point is set by the shape of the effective potential, and measured in each system’s own orbital period its rate is nearly the same for every pair of bodies.

Near the Sun–Earth points the e-folding takes twenty-three days instead of a day and a half, because the system’s natural clock is a year rather than a month. Everything above applies with that change of unit: the digits run out in a year and a half rather than two months, the basin shrinks over months rather than days, and the pieces of a multiple-shooting chain can be weeks long. That is why missions to the Sun–Earth points can be designed and flown with a more relaxed cadence than anything near the Moon — and why a transfer that links the two systems, lingering near a Sun–Earth point and then dropping towards the Moon, has to be designed in pieces whose lengths differ by an order of magnitude from one end to the other.

What the figures assume #

The amplification is drawn for a trajectory held at the Lagrange point itself, where the variational equations are exactly the linearisation. A real trajectory passes near the point for a while and then leaves, so its amplification grows exponentially for as long as it lingers and then slows; the number of e-foldings it accumulates is the time spent near the neck divided by the e-folding time, and that is the quantity the basin figure is drawn against. The three-body problem is planar and circular; the Moon’s eccentric, inclined orbit and the Sun’s pull on the Earth–Moon system change the rates by tens of per cent and add a dependence on the date. The basins are found for one starting offset and one direction of guess error; other directions give different basins, all shrinking at a similar rate.

Still open: finding the guess without a search #

The tubes supply the qualitative path, but turning a pair of tubes into a specific mission still involves a search over where along each tube to leave, how to match them in position and velocity, and how to fit the third and fourth bodies — the Sun’s pull on an Earth–Moon transfer, or the Moon’s on a Sun–Earth one — that the planar problem leaves out. For transfers that pass through several necks, or that link several three-body systems, the search space grows combinatorially, and methods that construct a good first guess directly from the dynamical structure, rather than by trying many, remain an active problem. The ordinary interplanetary trajectory is well conditioned enough for Newton’s method from a conic; the cheapest ones are the least, and the price of their cheapness is paid in the difficulty of finding them.