Aiming at a plane instead of at a planet

The rung below settled what classifies an open orbit: not the eccentricity but the hyperbolic excess speed, the residual velocity a body still has after the primary has finished pulling on it.

Getting a spacecraft to a particular open orbit is a different problem, and it has an answer that looks arbitrary until the reason is stated. A trajectory is not targeted by its closest approach distance, and it is not targeted by its periapsis latitude or its arrival time. It is targeted by a point in a plane — and the plane is defined by the trajectory itself.

What the plane is #

Take the incoming asymptote of the hyperbola: the straight line the spacecraft would follow if the planet had no gravity, running along the direction $\hat{S}$ of the incoming velocity $\mathbf{v}_\infty$.

The B-plane is the plane through the centre of the target body perpendicular to $\hat{S}$. The aim point $\mathbf{B}$ is the vector from the body’s centre to the point where that asymptote pierces it — which is the impact parameter of the encounter, the perpendicular distance from the centre to the undeflected path.

Two axes are laid out in the plane by convention. $\hat{T}$ is taken parallel to some reference plane, usually the target’s orbit plane or the ecliptic; $\hat{R} = \hat{S}\times\hat{T}$ completes the set. Every arrival is then quoted as a pair of numbers, $\mathbf{B}\cdot\hat{T}$ and $\mathbf{B}\cdot\hat{R}$, in kilometres.

Why not aim at the periapsis distance #

The obvious coordinate would be the closest approach distance $r_p$, which is what a mission actually cares about. The relation between it and the aim point is exact:

$$r_p = \frac{\mu}{v_\infty^2}\left(\sqrt{1 + \left(\frac{B v_\infty^2}{\mu}\right)^2} - 1\right).$$

Far from the planet, where $Bv_\infty^2/\mu \gg 1$, that reduces to $r_p \to B$: with focusing negligible, the aim point is the miss distance. Close in, where the same quantity is small, it reduces to $r_p \approx B^2 v_\infty^2/2\mu$ — quadratic in $B$.

So $r_p$ is a strongly nonlinear function of the thing a manoeuvre controls, and its sensitivity varies by a factor of several across any real targeting range. That is fatal for a linearised control problem: the partial derivative $\partial r_p/\partial \Delta v$ is not a constant, so a correction computed from one linearisation is wrong by the amount the derivative has changed since.

The aim point has no such problem. A small velocity change $\Delta\mathbf{v}$ applied a time $t$ before arrival displaces the asymptote by very nearly $\Delta \mathbf{v}\,t$ resolved into the plane, whatever $\mathbf{B}$ happens to be. The partial derivatives are essentially constant over the whole region of interest, and the targeting problem becomes a two-by-two linear solve.

A coordinate is chosen for the behaviour of its derivatives, not for its physical meaning, and that is the whole content of the B-plane.

The delivery ellipse #

Navigation does not deliver a point. It delivers a distribution, and the distribution is drawn in the B-plane as an ellipse: the projection of the trajectory’s covariance onto the two coordinates that matter.

The shape of that ellipse is informative in itself. It is usually long in one direction and short in the other, and the long axis points along the direction the tracking is worst at — which for a spacecraft tracked by Doppler alone is the direction across the line of sight. Adding an interferometric measurement rotates and shortens it, which is why deep-space navigation budgets pay for angular data even though the ranging is a thousand times more precise.

The manoeuvre schedule follows from the same arithmetic. A correction applied early is cheap, because a small velocity change has a long time to act: the displacement per unit $\Delta v$ is proportional to the time remaining. It is also imprecise, because the trajectory is not yet well known. A correction applied late is expensive and accurate. Every mission therefore has a sequence, each step smaller than the last, and the total propellant is dominated by the earliest one and the final accuracy by the latest.

What the aim point buys #

Three quite different things are all specified as a location in this plane, which is a large part of why it is used.

A turn. For a flyby whose purpose is to change the heliocentric orbit, what matters is the deflection angle, and

$$\tan\frac{\delta}{2} = \frac{\mu}{B v_\infty^2}.$$

Choosing the aim point chooses the turn. There is a hard limit — the planet has a surface, or an atmosphere — so the largest available turn corresponds to the smallest safe $B$, and everything a tour can achieve at a given encounter speed is bounded by it.

An entry corridor. For a probe that must survive an atmosphere, the requirement is a band of entry flight-path angles, and that band maps directly to an annulus in the B-plane. The Galileo probe’s corridor at Jupiter was a few tens of kilometres wide out of an aim point of tens of thousands, delivered from a release six months earlier. A prohibition. Planetary-protection rules for missions carrying unsterilised hardware past a possibly habitable moon are written as a probability of impact, and the calculation is exactly the one an impact probability for an asteroid uses: a distribution in the target plane, a capture cross-section, and an integral. Cassini’s end-of-mission disposal into Saturn was designed against a requirement of that shape — a bound on the chance that an uncontrolled spacecraft would eventually strike Enceladus or Titan, computed decades forward through a covariance.

Those three requirements pull in different directions, and reconciling them is most of what an arrival design consists of. A larger turn wants a smaller aim point; an entry corridor wants a specific narrow annulus; a prohibition wants the delivery ellipse to have no overlap with a forbidden region at all. All three are drawn on the same two axes, in kilometres, which is exactly why those axes are the ones everybody works in — a plane in which every requirement, every measurement and every manoeuvre can be drawn as a region is worth more than a plane with a tidier physical meaning.

How the aim point is known #

Nothing in this subject is measured directly, and the aim point is a good example of how far a quantity can be from the observation that fixes it.

A deep-space spacecraft is tracked in three ways, and none of them measures a position.

Two-way Doppler measures the rate of change of the distance to the tracking station, by comparing the frequency of a signal returned by the spacecraft’s transponder with the one sent to it. The precision is extraordinary — a few hundredths of a millimetre per second over a few minutes — but it is a rate, and along one direction only.

Ranging measures the round-trip light time of a coded signal, giving a distance good to about a metre at any distance. Also one-dimensional.

Delta-differential one-way ranging compares the arrival time of the spacecraft’s signal at two widely separated antennas against the arrival time of a quasar’s, which cancels most of the media and clock errors. It measures an angle, to a couple of nanoradians — which at five astronomical units is about a kilometre and a half.

The aim point is then a fitted parameter, in exactly the sense the rest of this field uses the word: an orbit determination takes weeks of those three data types, fits a trajectory, propagates it to the encounter, and reports where the asymptote lands. Nobody ever measures $\mathbf{B}$; it is the output of a solution, with a covariance, and the delivery ellipse is that covariance drawn.

The Earth’s own rotation is what makes the geometry work. A station’s line of sight to the spacecraft swings through a day, so a series of Doppler measurements over several hours samples the trajectory from a range of directions and recovers information about the two coordinates the instantaneous measurement cannot see. It is the same trick as an orbit determination from angles alone, with the roles of the observer and the target exchanged.

What that machinery has actually delivered #

The numbers are worth stating, because they are the argument for the whole apparatus.

Voyager 2 reached Neptune in August 1989 after twelve years and four and a half billion kilometres, and arrived within about thirty kilometres and a few seconds of its aim. New Horizons reached Pluto in July 2015 after nine and a half years, and arrived inside a target box seventy kilometres and a minute across — a requirement set by the need to point instruments at a body whose position was itself uncertain by hundreds of kilometres before the approach imaging began.

Neither number is a statement about propulsion. Both are statements about a fit: the trajectory was determined, the residual aim error was computed in the B-plane, a manoeuvre of a few centimetres per second was designed to remove it, and the process was repeated. The final corrections on both missions were smaller than the noise on a domestic bathroom scale expressed as a velocity.

Aiming deliberately at the wrong place #

There is a practice in arrival design that looks like an error and is the opposite of one.

A spacecraft carrying hardware that has not been sterilised must not impact a body that might be habitable, and the requirement is a probability rather than an intention. So the trajectory is not aimed at the target until quite late: it is biased, aimed at a point in the B-plane far enough from the body that even a total loss of control leaves the spacecraft missing.

The bias is removed in stages. As the tracking improves and the delivery ellipse shrinks, successive manoeuvres move the aim point closer to the intended one, and at each stage the aim is chosen so that the residual impact probability — computed from the current ellipse, not the eventual one — stays below the requirement.

That converts the manoeuvre sequence from a convergence into a negotiation. An early correction cannot simply target the final aim point even if it could reach it, because doing so would put an impact inside the current uncertainty; and a late correction cannot be deferred indefinitely, because the propellant needed grows as the time remaining falls.

The clearest applications are the Jupiter missions, where a flyby of Europa is scientifically valuable and an impact on it is prohibited, and the arrival design has to deliver a close pass from a trajectory that was, weeks earlier, aimed at empty space.

The same logic applies to the end of a mission. A spacecraft that has run out of propellant is a spacecraft whose trajectory can no longer be corrected, so its final state has to be one from which no plausible perturbation produces an impact for decades or centuries — which is a statement about a distribution in a target plane propagated forward through a chaotic multi-body system.

A requirement expressed as a probability is met by aiming somewhere the mission does not want to go, and unwinding that bias on a schedule is a substantial part of what an arrival campaign consists of.

The linearisation, and where it fails #

The whole apparatus rests on the assumption that the map from a small $\Delta \mathbf{v}$ to a displacement of $\mathbf{B}$ is linear. Three things break it.

A close approach on the way. If the trajectory passes another body between the manoeuvre and the arrival, the map is no longer linear — the intermediate encounter amplifies some directions and compresses others, sometimes by orders of magnitude. Multi-flyby tours are therefore targeted encounter by encounter rather than end to end.

An aim point close to the capture radius. The relation between $B$ and $r_p$ is steep there, so although $\mathbf{B}$ is still well behaved, the requirement expressed in $r_p$ becomes brutally tight. A mission aiming at a periapsis of one planet radius has to control $B$ far better than one aiming at ten.

A long lever arm. The displacement per unit $\Delta v$ grows with the time remaining, which sounds like an advantage and is also an error multiplier: an execution error in the manoeuvre is magnified by the same factor. Early manoeuvres therefore need clean-up manoeuvres, and the sequence in the figure above is not conservatism but arithmetic.

There is a fourth failure that is not a linearisation problem at all. The B-plane is defined by the incoming asymptote, and the incoming asymptote is a property of the trajectory being corrected — so the coordinate system moves when the trajectory does. For a small correction that is a second-order effect and ignorable. For a large one it is not, and the standard practice is to re-derive the plane after each manoeuvre rather than to keep targeting in the old one. A correction quoted in a stale B-plane is quoted in a frame that no longer exists, which is the sort of error that survives every check because both numbers are individually correct.

The bias is therefore a schedule as much as a geometry, and it is set at design time rather than negotiated in flight.

The same plane, for something nobody is steering #

The coordinate was built for spacecraft and it is used just as heavily for objects that are not being flown, and the transfer is nearly exact.

An asteroid approaching the Earth has an incoming asymptote, an excess speed and an aim point, and its uncertainty is a distribution in the same plane. The impact probability is the integral of that distribution over the region whose aim points lead to a strike — which, with focusing, is a disc of radius $R\sqrt{1 + v_{\text{esc}}^2/v_\infty^2}$ rather than the Earth’s own radius.

Two differences of practice are worth noting. For a spacecraft the distribution is nearly Gaussian, because the uncertainty comes from measurement noise on a well-modelled trajectory over a short interval. For an asteroid observed over years and propagated over decades, the uncertainty region is stretched and curved by the nonlinear dynamics, so the “ellipse” is a long thin arc and integrating over it requires sampling rather than a covariance.

And the asteroid case has a second target plane in use — one defined not by the object’s own asymptote but by the perpendicular to the Earth’s velocity, which is more convenient when the encounter is slow and the focusing is strong. Reports quoting a miss distance in one plane and an uncertainty in the other are a known source of confusion.

What survives unchanged is the reason for using a plane at all. An impact probability is an integral over a region, and a region is only well defined in coordinates where the dynamics are approximately linear — so the argument that made the B-plane the right choice for aiming a spacecraft makes it the right choice for deciding whether something is going to hit.

The plane is the same construction at every target, and the two things that change between targets are worth drawing.

Where this ladder goes next #

This rung has established the targeting coordinate and why it is the one used: linear in the control, exactly related to the turn, and the natural home for a delivery ellipse.

The rung above is the tour, where the aim point at one encounter determines the arrival conditions at the next, and where the design problem becomes a search over sequences of B-plane targets. What one flyby can reach is bounded by the invariant it cannot change, and the aim point is the only free variable inside that bound.

Beside it lies the capture problem: what has to be given up, in propellant or in atmosphere, to convert a hyperbolic arrival into a bound orbit — and why the cheapest place to do it is the periapsis the aim point has just chosen.

And below it, the habit this rung is really about: the right coordinate is the one whose derivatives are constant. Aiming at the quantity a mission cares about is natural, comfortable, and the reason a targeting problem becomes ill-conditioned.