The relative orbit that breathes

A burn that moves the wrong way established the frame that a close approach is flown in — after the phasing problem had been settled — and three results in it: a free relative orbit is an ellipse exactly twice as long along the track as across it, a prograde burn leaves a chaser eight kilometres behind after one revolution, and a radial burn returns it exactly to where it started.

Every one of those is a property of a circular target orbit, and the dependence is not a technicality. The linearised equations have constant coefficients — the mean motion appears as a number rather than as a function — and constant coefficients are where the whole of the tidiness comes from.

An eccentric target has no constant mean motion. The frame rotates at a rate that varies through the orbit, the gravity gradient varies with it, and the elegant results go.

What actually changes #

Writing the relative equations about an eccentric reference gives

$$\ddot{x} - 2\dot{\theta}\dot{y} - \ddot{\theta}y - \dot{\theta}^2x = -\frac{\mu}{r^3}x \cdot(\text{terms}), \qquad \ddot{y} + 2\dot{\theta}\dot{x} + \ddot{\theta}x - \dot{\theta}^2 y = \dots$$

and the important feature is not the extra terms but the fact that $\dot\theta$ and $r$ are now functions of time. Two things follow immediately.

The coefficients are periodic rather than constant. So there is no single characteristic frequency, and the notion of “a free relative orbit” as one shape traversed once per revolution is lost. What replaces it is a family of paths that repeat once per revolution of the target and that change shape continuously through it.

And there is an extra torque term. The $\ddot\theta$ terms exist only because the frame’s rotation rate is changing, and they are what makes a radial displacement couple to an along-track one in a way that depends on where in the orbit the vehicle is.

The equations are still linear, which is more than one might expect, and they still have a closed-form solution — but only after changing the independent variable from time to the target’s true anomaly, at which point the system becomes linear with periodic coefficients and a state transition matrix can be written down. That transformation is the central trick of the eccentric problem, and it was found in 1965 and given a clean non-singular form in 2002.

Forty to one, in the frame’s own rotation #

The abstraction is easier to believe with a case in it, and the standard one is a satellite stranded where a launch left it.

A geostationary transfer orbit has a perigee about 250 kilometres up and an apogee at 35,786, so its radius runs from 6,628 kilometres to 42,164 — a factor of 6.4. Equal areas are swept in equal times, so the angular rate goes as the inverse square of the radius, and the frame a proximity operation is flown in turns forty times faster at perigee than at apogee.

Everything about a close approach scales with that rate. The Coriolis terms carry a factor of it; the along-track distance corresponding to a given angular displacement carries a factor of its inverse square; the time available between two waypoints carries its inverse. A procedure written for the apogee end of the orbit, executed near perigee, is not a procedure that costs a little more — it is one whose every characteristic time is forty times shorter.

The vehicle also spends almost none of its time at perigee. By the same law, the fraction of a period spent inside a given radius is heavily weighted toward apogee, so the fast part of the orbit is a few per cent of the clock and the slow part is most of it. Approach operations are therefore scheduled entirely in the apogee half, not because the dynamics there are better understood but because there is time.

The one thing that survives #

The circular theory’s most useful statement is not about shape at all, and it is exactly the one that carries over.

A relative orbit drifts secularly if and only if the two semi-major axes differ, and the drift rate is $-\tfrac{3}{2}n\,\Delta a$, which is the derivative of the law linking period to size and nothing else. Two orbits with the same semi-major axis have the same period; two bodies on them return to the same relative configuration every revolution, whatever their eccentricities, inclinations or phases.

That is a statement about energy, since the semi-major axis and the specific orbital energy are the same quantity, and it needs no linearisation, no small separation and no circular reference. Every path in both figures above closes exactly, and the reason is that one number was held fixed.

So the practical content inverts. Around a circular target, a mission plans a shape: a safety ellipse of a chosen size, traversed once per orbit. Around an eccentric one, a mission plans an energy: match the semi-major axis, and accept whatever shape the geometry then produces — which cannot be chosen and has to be computed.

The cost of a manoeuvre depends on when it is flown #

The consequence that matters operationally is the loss of time-invariance.

Around a circular target the state transition matrix depends only on the elapsed time, so a two-impulse transfer from one relative state to another costs the same wherever in the orbit it begins. A procedure written once is a procedure that can be executed at any time, and the constraint on scheduling is the lighting and the ground coverage rather than the dynamics.

Around an eccentric target the matrix depends on the true anomaly at the start and at the end, so the same transfer between the same two relative states costs different amounts at different points of the orbit — by a large factor when the eccentricity is large. The reason is visible in the second figure: near periapsis the frame is sweeping round quickly and a relative displacement is being carried through a large angle in a short time, which takes a bigger impulse to arrest.

Three operational consequences follow, and all three are why eccentric-target rendezvous is a separate discipline rather than a correction.

The approach is timed rather than merely sequenced. Braking gates are placed at chosen true anomalies, not at chosen ranges, and a hold that overruns is not a delay but a different manoeuvre.

Passive safety has to be recomputed continuously. A safety ellipse around a circular target is safe forever; around an eccentric one the closed path changes shape through the orbit, and whether it passes through the target’s position depends on where in the orbit the vehicle was placed on it.

And the linearisation fails sooner. The expansion parameter is the separation divided by the instantaneous orbital radius, which for an eccentric target is smallest at periapsis and largest at apoapsis — the same accounting a geostationary satellite’s station-keeping budget is written in, applied to a reference that will not hold still — so a separation that is comfortably linear at one end of the orbit is not at the other.

The variable that makes it linear #

The eccentric problem has a closed-form solution and finding it required giving up something that looks non-negotiable: time as the independent variable.

Written against time, the relative equations have coefficients that are functions of time through the target’s radius and angular rate, and there is no general closed form for a linear system with arbitrary time-varying coefficients. Written against the target’s true anomaly instead, and with the relative coordinates scaled by the target’s own radius, the same system becomes linear with coefficients that are simple functions of the new variable — and it integrates.

The cost is that the answer arrives as a function of where the target is in its orbit rather than of how long has elapsed. Converting back requires solving Kepler’s equation, which is a separate numerical problem and the reason the eccentric solution is described as closed-form and used as a subroutine.

The move is a familiar one in a different corner of the same subject. The singularity in a two-body propagation that turns out to be a change of variable is the same instinct: a problem whose difficulty lives in a coordinate is a problem to re-coordinate rather than to approximate. Choose the independent variable that makes the coefficients constant, and if none does, choose the one that makes them simple.

There is a practical reason this matters beyond elegance. A guidance algorithm running on a spacecraft cannot integrate a stiff system in real time and can evaluate a matrix. The value of a closed form here is not that it is exact — a numerical integration would be exact enough — but that it is cheap, and cheapness is what allows a vehicle to re-plan its own approach between one waypoint and the next.

Where the eccentric case is actually flown #

The circular theory has served for sixty years because the targets have been nearly circular. The International Space Station’s orbit has an eccentricity of about five ten-thousandths, which makes its orbital radius vary by three kilometres out of six thousand seven hundred — well below the level at which any of this matters.

Four classes of mission do not have that luxury.

Servicing a satellite in a transfer orbit. A vehicle stranded in geostationary transfer is on an orbit of eccentricity 0.73, and reaching it means flying proximity operations where the relative dynamics change by an order of magnitude between perigee and apogee. This is the case the modern eccentric-rendezvous literature was written for.

Highly elliptical operational orbits. A spacecraft in a twelve-hour orbit of eccentricity 0.7, of which several constellations exist, has the same problem for any rendezvous with it.

Small bodies. An asteroid or comet is on a heliocentric orbit of substantial eccentricity, and a spacecraft station-keeping near one is in exactly the configuration of the second figure — although for a body small enough, its own gravity and the pressure of sunlight are larger effects than the solar tide, so the problem changes character rather than merely becoming harder.

And debris removal. The objects most worth removing are in orbits nobody chose, and a defunct upper stage that failed during a burn is typically in an eccentric one — which the orbit that has to be paid for every year never is. This is the case that makes the subject current: the rendezvous that has to be flown is with a target that is uncooperative, tumbling and in a bad orbit, and each of those three is a separate difficulty.

What was actually measured #

Nothing on these figures is an observation. Both are exact propagations of the two-body problem — a Kepler equation solved by Newton’s method, positions differenced, and the difference rotated into the target’s own frame.

That choice is deliberate and it is the check. The earlier essay draws the linearised solution and verifies it against an integration of the linearised equations, which establishes that the closed form is right and says nothing about whether the linearisation is. Here the propagation is of the real equations, and the fact that it returns the two-to-one ellipse at zero eccentricity is a check of the linearised account rather than of this one.

What is measured, in a real operation, is a relative state: range and range rate from a radar or a laser, two bearings from a camera, or a full relative position from differential satellite navigation. The dynamics above are what converts a sparse sequence of those into a continuous estimate, and an error in the dynamics model shows up as a filter that has to be retuned rather than as an obviously wrong number.

Where the model stops #

Two bodies and no more. Everything here is two Keplerian orbits about a point mass. The Earth’s oblateness makes the two vehicles’ nodes and perigees precess at slightly different rates if their orbits differ at all, and over more than a few orbits that is the dominant term. It is also, unlike the eccentricity, a secular effect: it accumulates rather than repeating, so it cannot be absorbed into a periodic solution.

The separation is still small. Propagating exactly removes the linearisation error in the dynamics and not the assumption that the two vehicles are near each other, which enters through the whole idea of a local frame. At a separation approaching the orbital radius the notion of a radial and an along-track coordinate stops being useful.

And nothing here is three-dimensional. The out-of-plane motion decouples exactly around a circular target — that is one of the circular theory’s cleanest results — and around an eccentric one it does not. A cross-track offset couples into the in-plane motion through the varying rotation rate, which is a genuine complication and is the one this essay’s figures cannot draw.

The chaser is a point. A tumbling target has a body-fixed docking port whose position in the relative frame is the sum of the orbital motion and the rotation, and past a certain point the orbital part becomes the smaller of the two.

What a safety ellipse becomes #

That same essay names passive safety as the reason an approach is flown from below, and it is worth following that idea into the eccentric case because it is where the difference stops being abstract.

A passively safe configuration is one in which a total loss of control does not produce a collision: the vehicle continues on its free relative orbit, and that orbit never passes through the target. Around a circular target this is a design problem with a clean answer. Every free relative orbit is the same ellipse, a cross-track oscillation of the right phase tilts it into a three-dimensional loop whose projection never crosses the origin, and the minimum separation it guarantees is a number chosen once.

Around an eccentric target none of that holds. The free relative orbit changes shape through the revolution, so the minimum separation it reaches is not a property of the configuration but of the configuration and the point in the target’s orbit at which it was entered. Two vehicles placed on nominally identical safety ellipses at different true anomalies have different closest approaches, and the difference can be large.

So the quantity a mission has to guarantee — that the free trajectory clears the target by at least some margin, for at least some number of revolutions, from the state the vehicle is in now — becomes a computation that has to be redone continuously rather than a box drawn in advance. Autonomous vehicles carry that computation on board, and the interesting design question is how far ahead it has to look: too short and a slow drift is missed, too long and the linearisation it uses has stopped being valid.

The safety property is the same and the object certifying it is not. A certificate that was a geometric statement about a shape becomes a numerical statement about a trajectory, and a numerical certificate has to be recomputed whenever anything changes — which, on an eccentric orbit, is continuously.

The generalisation #

The shape worth carrying is about what a special case is hiding.

The circular theory is not an approximation to the eccentric one in the ordinary sense of being a first term. It is a degenerate case: setting the eccentricity to zero makes a set of periodic coefficients constant, and constant-coefficient systems have properties — a characteristic frequency, time-invariance, a single fixed shape for the free solutions — that periodic-coefficient systems simply do not have. The results that vanish do not vanish gradually; they stop being the kind of statement that can be made.

A parameter that appears in a coefficient rather than in a term is the one to check first. A small term added to an equation perturbs its solution; a small variation in a coefficient can change what kind of solution the equation has. That is why an eccentricity of 0.0005 at the space station is genuinely negligible and an eccentricity of 0.1 is a different problem rather than a ten per cent correction.

The second reading is the more useful one for anybody designing something. When the geometry of a problem stops being computable in advance, look for the conserved quantity underneath it — because a statement about a conserved quantity survives the loss of the geometry. Here the shape went and $\Delta a = 0$ stayed, and a mission that plans around the energy rather than around the picture is a mission that does not have to be redesigned when the reference orbit changes.

Still open: the measurement the dynamics are fed #

What comes next is the other half of a rendezvous, which this essay has taken for granted: the relative state has to come from somewhere, and what a spacecraft’s own sensors deliver is not a position. A camera measures two angles and no distance, and the linearised dynamics that make everything above tractable have a property that turns out to be fatal for that particular instrument.

Beside it lies the guidance problem proper — choosing a sequence of impulses of minimum total size subject to staying outside an approach corridor and remaining passively safe throughout — which around a circular target is a linear program with a closed-form answer and around an eccentric one is a linear program whose constraints move.