A correction that tilts the arc it meant to aim

Lambert’s problem asks for the orbit that carries a body from one point to another in a given time, and in a field with one centre of attraction it has an exact answer. The Earth’s field does not have one centre. Its equatorial bulge adds a force a thousandth of the central one that turns orbital planes and lines of apsides, and a velocity solved from the two-body problem and flown in the real field arrives kilometres to a hundred kilometres from where it was aimed.

The repair used in practice needs no theory of the perturbation at all. Fly the two-body solution through the real field, measure where it arrives, and solve Lambert’s problem again toward a target displaced by the opposite of the miss; fly that, measure, and repeat. Each pass removes most of the remaining error. For a transfer of a third of an orbit in low Earth orbit the miss shrinks by a factor of a few hundred to a few thousand with each pass, and three passes take it from tens of kilometres to millimetres.

The hero figure shows the factor by which each pass shrinks the miss, across transfer angles from 120° to 240°, for a transfer between orbits at 400 and 420 kilometres inclined at 51.6°, the inclination of the International Space Station. At the bulge’s real strength the factor is a few thousand at 150° and falls toward 180°: by 178° it is 83, by 181° it is 8. The iteration always converges, but near the straight-through geometry it is a hundred times slower. With a perturbation ten times stronger it fails outright between 181° and 184°; with thirty times, between 182.5° and 192°.

Why the corrected target works at all #

The corrected-target iteration is a fixed-point iteration. The miss is a function of the aim point; the iteration moves the aim point by minus the miss. It converges if moving the aim point by some distance changes the miss by a smaller distance — if the map from aim to miss is a contraction.

Away from 180° that is easy to satisfy. The bulge’s effect on the transfer depends on the transfer’s geometry: the latitudes it crosses, how long it spends near the equator, its plane. Moving the aim point by a few tens of kilometres changes that geometry very little, so the miss it produces changes very little. Each pass moves the aim almost exactly to where it should be, and what is left is the small change in the perturbation caused by the change in aim — a second-order effect, which is why the miss falls by orders of magnitude per pass.

That argument has one weak point. It assumes that a small change in the aim point is a small change in the geometry. For most transfers it is. For one family it is not.

A kilometre sideways is a degree of tilt #

A Lambert transfer lies in the plane containing the two end points and the centre of the Earth. When the transfer angle is 90°, those three points define the plane robustly: moving one end point a kilometre out of the plane tilts it by about a kilometre divided by the radius, a hundredth of a degree. When the transfer angle approaches 180°, the end points and the centre come into line, and a line does not define a plane.

At 179° a kilometre of sideways aim tilts the transfer plane by half a degree. The bulge’s miss has a sideways component — the bulge turns the orbit’s plane, and an arc flown for half an orbit arrives displaced out of the plane it was designed in. Correcting that sideways miss moves the aim sideways, which near 180° rotates the whole arc onto a new plane, which crosses different latitudes and feels the bulge differently, which changes the sideways miss. The correction feeds back into the error it is correcting, with a gain that grows as one over the sine of the transfer angle.

This is the same singularity that puts a wall across the porkchop plot of interplanetary transfers. There, transfers to Mars that go almost exactly half-way round the Sun are forced into steeply inclined planes, because the planes are set by the planets’ small departures from a common plane, and the wall sells inclination at the dearest price. Here the same geometry appears in an iteration rather than in a cost: near 180° the plane is cheap to change and expensive to hold still.

Three iterations, one that runs away #

The figure that makes the failure concrete follows the miss through successive passes.

The strengthened bulge stands in for what a real mission would meet over a longer transfer: the displacement the bulge produces grows roughly in proportion to the time it acts, so a transfer of several revolutions, or one in a lower and more strongly perturbed orbit, reaches misses of this size. The three angles are only eleven degrees apart. At 170° the iteration is still a strong contraction; at 178° it has slowed to a crawl, losing less than half the miss per pass after the first; at 181° it expands, each pass tilting the arc further out of the plane it was meant to stay in.

The asymmetry about 180° in the figures — failure sets in just beyond it rather than just before it — comes from the direction in which the bulge pushes the arc sideways relative to the direction a sideways correction tilts it, and depends on the inclination and the direction of travel. The symmetric part, which is the important one, is the one-over-sine growth of the gain on both sides.

An equatorial arc has no plane to lose #

The diagnosis makes a prediction that can be checked directly. If the slowing comes from the bulge pushing the arc out of its plane, an arc the bulge never pushes sideways should not slow at all. An equatorial orbit is such an arc: the bulge is symmetric about the equator, so a body moving in the equatorial plane feels a stronger inward pull but no force out of the plane, and its plane never turns.

The equatorial transfer converges at the same rate at 150°, 178° and 181°. The singular geometry is still there — the end points and the centre still come into line — but the iteration never asks it a sideways question, because no correction is ever sideways. The slowing near 180° is therefore not a property of Lambert’s problem or of the iteration in general. It is a property of an iteration that corrects a sideways error through a geometry that cannot hold its plane.

A polar orbit, the other extreme, is worse than the station’s. Run through the same calculation at 98°, the inclination of the sun-synchronous orbits used for Earth observation, the iteration shrinks the miss only sixfold per pass at 178° at the real bulge, and with ten times the bulge it fails across six degrees, from 175.5° to 181.5°. How hard the bulge pushes an arc sideways depends on where the arc crosses the equator and how steeply, and the polar arcs that carry most remote-sensing satellites are pushed hardest in the sense that matters here.

A cone that widens with the perturbation #

The size of the region where the iteration fails depends on how large the perturbation is.

The cone opens when the perturbation’s sideways miss, multiplied by the tilt per kilometre, becomes comparable to the miss itself, and since the tilt per kilometre goes as $1/\sin\theta$ the cone’s half-width grows roughly in proportion to the perturbation. At the Earth’s real bulge, for a transfer of less than an orbit, the cone is narrower than a degree and in practice never met: the gain near 180° is large but still below one. For transfers of many revolutions, for orbits much lower than the station’s, or for transfers around bodies more oblate than the Earth, the cone opens to degrees or tens of degrees and swallows a useful range of geometries.

Correcting only within the plane #

The diagnosis suggests the cure. The iteration fails because it steers with the one coordinate that rotates the arc. If the sideways part of each correction is discarded — if the aim point is moved only within the transfer’s original plane — the arc cannot be rotated, and the feedback is cut.

With the in-plane correction the iteration converges in a pass or two at both angles, including the one where the full correction ran away. What it converges to is not a hit: it is an arrival in the right place within the plane and displaced sideways by whatever the bulge did to the plane. That sideways miss is removed separately, by a small plane change at arrival — or, better, by turning the plane where turning is cheap, at the node of the two planes rather than at the arrival point.

The cost is real: at ten times the bulge the plane change is about a hundred and fifty metres a second; at the real bulge it is about a tenth of that. But it is a cost that can be calculated and budgeted, where the alternative is an iteration that does not return an answer at all. The trade is between solving the problem as posed — hit this point at this time with one impulse — and solving a nearby problem that has a robust solution.

The commonest transfer sits on the singularity #

There is an irony in where the failure lives. The 180° transfer is not an exotic geometry. It is the Hohmann transfer, the cheapest two-impulse route between circular orbits, and it goes exactly half-way round by construction: depart at periapsis, arrive at apoapsis, 180° later. Half-orbit transfers are the backbone of orbit raising, of the phasing manoeuvres that bring a spacecraft up behind a space station, and of the burns that move a chaser along the track during a rendezvous.

A coplanar Hohmann transfer has no difficulty with its plane, because the plane is given — both orbits share it — and nobody asks Lambert’s problem to find it. The difficulty appears when a near-half-orbit transfer is targeted point to point in a perturbed field, which is exactly what rendezvous guidance does in its final hours: it aims at where the target will be, with the arrival time fixed, and corrects for the bulge by iteration. Point-to-point targeting near half an orbit is known in that field to be ill-conditioned, and the usual responses are the two the figures suggest: choose a transfer angle comfortably away from 180°, or treat the out-of-plane part of the problem separately from the in-plane part. The terminal phase of a crewed rendezvous has traditionally been flown as a transfer of well under 180°, for this and for other reasons — lighting, the chaser’s view of the target, the safety of the approach if a burn fails.

So the in-plane correction drawn above is not a new idea. It is the formal version of a habit: hold the plane fixed by fiat near 180°, solve the problem in the plane, and fix the plane with a separate small burn. What the figures add is the reason the habit is necessary, and a way to see in advance where it becomes necessary for a given orbit, perturbation and transfer.

What the iteration is really doing #

The corrected-target iteration is a special case of a more general method, differential correction: adjust the free parameters of a trajectory to drive its errors to zero, using the sensitivity of the errors to the parameters. The full method computes that sensitivity — the matrix of partial derivatives of arrival position with respect to aim or to departure velocity — and solves for the adjustment, which is Newton’s method in many dimensions. The corrected target replaces the matrix with the identity: it assumes that moving the aim moves the arrival by the same amount.

Near 180° that assumption fails, and Newton’s method with the true matrix would not: it would see the large sensitivity to sideways aim and take a correspondingly small sideways step. Its cost is the matrix, which needs either the variational equations integrated alongside the trajectory or several extra trajectories flown with perturbed aims. The corrected target’s appeal is that it needs one trajectory per pass and no derivatives. The in-plane variant keeps that appeal by building in the one piece of the matrix that matters — the knowledge that the sideways direction is special near 180° — without computing the rest.

Ill-conditioned is not the same as unsolvable #

It is worth separating two things the figures might seem to conflate. The perturbed problem near 180° has a perfectly good solution: there is a departure velocity that, flown through the real field, arrives exactly at the target at the specified time. Nothing about the bulge makes it impossible. What fails is one particular way of finding it, the one that assumes the arrival moves as the aim moves.

The difference shows in what a Newton step would do. The derivative of arrival position with respect to aim point is close to the identity in the two directions within the plane and far from it in the sideways direction, where a small change of aim produces a large change of plane and so a large change of arrival. A matrix with one direction stretched like that is ill-conditioned: its inverse amplifies errors in the stretched direction, and a method that uses it must be careful about how precisely it knows the sideways miss. But it is invertible, and Newton’s method with that matrix takes a small, correct sideways step where the corrected target takes a large, wrong one.

The in-plane correction sits between the two. It uses the identity where the identity is right — within the plane — and refuses to move at all in the direction where the identity is wrong. That refusal costs it the ability to hit the target in one impulse, which is why it leaves a sideways miss, but it never moves in the wrong direction, which is why it always converges. A method that knew the sideways derivative would do better; a method that assumed it was one would do worse; a method that admits it does not know it and stays out of that direction does well enough to be the one used.

Where the model stops #

The perturbation is the bulge alone, scaled. Real transfers also feel the higher harmonics of the field, drag, and the Moon and Sun. Scaling the bulge stands in for a larger perturbation of the same character — one that turns the plane — and the iteration’s behaviour depends on that character: a perturbation acting purely in the plane, like drag, would not feed back through the plane’s tilt.

The transfer time is set by the mean motion. Each transfer takes the time a circular orbit would take to sweep its angle. Faster or slower transfers change the arc’s shape and the bulge’s effect on it, but not the geometric singularity at 180°.

The contraction ratio is measured from the first passes. Once the iteration is close to a fixed point its ratio approaches the gain of the linearised map; the figures take the smaller of the second and third passes’ ratios, which is that gain to within the noise of a few passes.

Exactly 180° is excluded. The plane is undefined there in the two-body problem itself, before any perturbation; a transfer of exactly 180° has to be specified with its plane given separately, which is what the in-plane correction does implicitly.

Still open: transfers of many revolutions #

Every transfer here goes less than once round. Rendezvous and constellation-maintenance problems often call for transfers of many revolutions, which Lambert’s problem admits as a family of solutions for each revolution count, and in which the bulge’s effect accumulates over every orbit flown. The plane’s sensitivity is still $1/\sin\theta$, but $\theta$ is now the transfer angle modulo a full turn, and the perturbation that has to be corrected is many times larger.

Whether the corrected target, or its in-plane variant, converges for such transfers — and how wide the cone of failure is around every half-integer number of revolutions — is the question that decides whether the method can be used where it would save the most work. A multi-revolution solver flown through the same field would answer it, and the answer would map the transfer angles and revolution counts for which a one-trajectory-per-pass correction is enough and those for which only the full derivative matrix will do.