A millimetre a second at launch, seventy-six kilometres at Mars

A trajectory to another planet is designed as a chain of simple pieces. Three two-body problems are stitched together: a hyperbola away from the Earth, an ellipse around the Sun, a hyperbola into the target’s gravity. Each is solved exactly, and the seams between them are where the method is knowingly false — at the surface where the switch from one attracting body to another is made, both neglected forces are at their largest, and the stitched path has a kink no real trajectory has.

The patched conic is not what is flown. It is the first guess for a calculation that finds what is flown, and that calculation — differential correction — is the subject here. It rests on one matrix, and the matrix rests on a single striking number.

Seventy-six kilometres for a millimetre a second #

The figure is a numerical integration. A craft leaves the Earth’s orbit at 1 astronomical unit on the Hohmann transfer to Mars’s orbit at 1.524, and its path is integrated under the Sun’s gravity for the 259 days of the transfer. Then the integration is repeated with the departure speed changed by a small amount, and the change in the arrival position is measured.

An error of one millimetre a second along the direction of motion moves the arrival by 76 kilometres. An error across the direction of motion moves it by 23. The relation is linear across five decades: ten millimetres a second is 760 kilometres, a metre a second is 76,000 — more than twenty times Mars’s radius. The best launch vehicles place a spacecraft on its departure hyperbola to within a few metres a second, and the atmosphere of Mars, which a lander must enter within a corridor a few kilometres wide at the right angle, is a target a thousand times smaller than that error. No interplanetary trajectory is flown as launched.

The number is not special to Mars. It is roughly the flight time multiplied by a factor of order a few, because an error in speed at departure is an error in the orbit’s energy, and an error in energy becomes an error in timing that grows for as long as the flight lasts.

An energy error becomes a timing error #

The way the deviation develops shows where the sensitivity comes from.

A craft launched slightly faster than planned is on a slightly larger orbit. It reaches slightly further from the Sun, so its radial position differs from the nominal by an amount that grows and oscillates with the shape of the orbit. More importantly, a larger orbit has a longer period, so the faster craft moves more slowly along the latter part of its path and falls behind the schedule it was meant to keep. That along-track difference is not an oscillation; it is an accumulating error in when the craft reaches each point, and it grows roughly in proportion to the time elapsed. By arrival, a hundred millimetres a second has become more than seven thousand kilometres of lateness.

The same thing happens to every orbit: a small change in a satellite’s energy becomes a drift along its track, which is why rendezvous and station-keeping are timing problems before they are position problems. The sensitivity at arrival is the accumulated result, and it is large simply because the flight is long.

Where seventy-six comes from #

The number can be estimated without an integrator, and the estimate shows which physics sets it. A small change $\delta v$ in the speed at perihelion changes the orbit’s energy, and through the vis-viva relation its semi-major axis, by

$$\frac{\delta a}{a} = \frac{2 a v\,\delta v}{\mu},$$

where $v$ is the speed and $\mu$ the Sun’s gravitational parameter. By Kepler’s third law the mean motion changes by $\delta n/n = -\tfrac32\,\delta a/a$, so the craft’s angular position along its orbit drifts from the nominal at a rate proportional to the elapsed time, and after a time $t$ the along-track displacement is roughly $a\,\delta n\,t$. Putting in the Hohmann transfer’s numbers — a semi-major axis of 1.26 astronomical units, a perihelion speed of 32.7 kilometres a second, half an orbit’s flight — gives about ninety kilometres per millimetre a second at Mars’s distance, within a fifth of the integrated 76. The rest is geometry: the faster craft’s orbit is also slightly more eccentric, which speeds it through the first part of the flight and partly offsets the lag it builds up later, and the estimate’s uniform drift ignores that.

The estimate makes two things explicit. The sensitivity grows linearly with the flight time, so a journey to Jupiter, four times longer, is four times more sensitive per unit of departure error before any flyby is considered. And it grows with the speed at departure, because a faster craft’s energy changes more for the same small change of speed. A millimetre a second of launch error is a fixed fraction of the orbital energy, and a fixed fraction of the energy becomes a fixed fraction of the period, which becomes a distance at the rate the craft is moving.

The matrix, and Newton’s method on it #

The linear relation in the first figure — the change in arrival position per unit change in departure velocity — is a matrix: two numbers for the position’s two components, for each of the velocity’s two components. It is one block of the state transition matrix, which maps a small change in the full state at one time into the change it causes at another, and it can be computed either by integrating the equations of motion’s linearisation alongside the trajectory or, as here, by integrating a few slightly different trajectories and differencing them.

With it, aiming is a linear problem. If a trajectory integrated from some departure velocity arrives a vector $\delta\mathbf r$ away from the aim point, the correction to the departure velocity is

$$\delta\mathbf v = -\,\Phi_{rv}^{-1}\,\delta\mathbf r,$$

where $\Phi_{rv}$ is the matrix. The correction is exact only for a linear problem, and the orbit is not linear, so the corrected trajectory misses too, by less. Repeat: integrate, measure the miss, recompute the matrix, solve, correct. That is Newton’s method, applied to the function that maps a departure velocity to an arrival position.

The convergence is dramatic. A departure velocity two metres a second wrong misses by 151 thousand kilometres. One correction brings it to 35 kilometres; a second, to six millimetres. The number of correct digits roughly doubles at each step — five, then eleven — which is the signature of Newton’s method close to a solution: the error after each step is proportional to the square of the error before it. After two steps the miss is at the level of the numerical integration’s own precision and stops improving, because there is nothing left to correct that the integration can represent.

A sketch made real in two steps #

The same procedure turns the simple model into the full one, and that is its main use.

A trajectory designed in the two-body problem — the Sun alone — and then integrated with Jupiter’s gravity added misses Mars by 36 thousand kilometres. Jupiter never comes closer than about three and a half astronomical units during the flight, and it still bends a 259-day trajectory by several times Mars’s radius. Differential correction, run with the matrix computed in the model that includes Jupiter, finds the departure velocity that hits the target in that model: two iterations, and a change of about seven-tenths of a metre a second.

The same holds for every force the simple model omits. The Earth’s gravity during the first days of the flight bends the departure far more than Jupiter bends the cruise; the pressure of sunlight on the craft’s solar panels pushes it by a few hundred kilometres over a transfer; each is a term in the force model, and each is absorbed by the correction without being designed for separately. That is how every interplanetary trajectory is actually built. The patched conic gives a departure velocity that is roughly right; differential correction in a model with every planet, the Earth’s and the target’s gravity, solar radiation pressure on the craft and relativistic corrections finds the one that is exactly right in that model. The kink the patched conic hides at the sphere of influence is a few metres a second — large for navigation and negligible for design — and it simply disappears in the correction, which never knew there was a seam. The simple model’s job is to put Newton’s method close enough to converge.

The target is usually not a point but a plane. At arrival, what matters is not where the craft is at a particular instant but where its approach hyperbola pierces a plane through the planet perpendicular to the incoming direction — the B-plane — which fixes the closest approach and the inclination of the arrival, with the arrival time a separate, third coordinate. The same differential correction aims at two coordinates in that plane, with a matrix that maps departure velocity into them.

Why correction burns come early #

A trajectory flown from a real launch is corrected in flight, and the question of when to correct has a clear answer.

A departure error of one metre a second can be cancelled on the second day of the flight by a burn of about one metre a second. Leave it until halfway and the burn must be five times larger; leave it until the last three weeks and it costs almost fifty. The miss grows in proportion to the time since departure, as the second figure showed, while the leverage a burn has on the arrival — the same matrix, from the burn to the arrival — shrinks in proportion to the time remaining. A late correction must move a large error with a short lever.

There is a further reason the first correction comes early, and it is deliberate error. The upper stage of the launch vehicle travels on nearly the same trajectory as the spacecraft, and unlike the spacecraft it has not been cleaned to the standard required of anything that might reach Mars. So the launch is aimed not at Mars but some hundreds of thousands of kilometres away from it, far enough that the stage cannot hit the planet by chance, and the spacecraft’s first correction burn removes the bias along with the launch error. The largest correction of the flight is planned before launch, and it is made early because early is cheap.

So every interplanetary mission plans its first trajectory correction manoeuvre within days to weeks of launch, to remove the launch vehicle’s error while it is cheap, and schedules later ones to remove what remains as tracking refines the orbit: each is smaller than the last, and the final ones, days before arrival, are centimetres a second, aimed at a corridor a few kilometres wide. The tracking that measures the errors — Doppler shifts to a small fraction of a millimetre a second, ranges to metres — is what makes the corrections possible, and the state transition matrix is what turns each new measurement into a burn.

When a flyby multiplies the error #

Not every trajectory is as well conditioned as a direct transfer. A trajectory that uses a planet’s gravity to change its course passes close to that planet, and the direction in which it leaves depends on how close it came. An error of a kilometre in the closest approach to Venus or the Earth becomes an error of many kilometres in the outgoing direction, and that direction error grows along the next leg exactly as a departure error does. Each flyby multiplies the sensitivity by a factor that can be tens or hundreds, and a trajectory with several encounters with the same planet or with several planets in sequence can have a matrix, from its first burn to its last target, with entries of millions of kilometres per metre a second.

Such trajectories are not flown by aiming once. They are flown by aiming at the next flyby, correcting after it, and aiming at the one after, with the matrix computed leg by leg so that each correction works against a manageable sensitivity rather than the product of all of them. Newton’s method on the whole trajectory at once would still converge in principle; in practice, with the matrix that ill conditioned, the first guess must already be extraordinarily good, and the design process builds it up one leg at a time.

At the extreme the sensitivity stops being large and becomes unbounded in the sense that matters. A trajectory in a chaotic region has a matrix whose entries grow exponentially with time rather than linearly, and past a few Lyapunov times no achievable departure precision determines the arrival at all. Interplanetary transfers are far from that regime; some of the low-energy transfers mentioned at the end are not.

The same matrix, run the other way #

Differential correction has a twin that is older and in some ways more important. Aiming asks what departure velocity produces a desired arrival; orbit determination asks what state at some epoch produces the observations actually made. Both are solved by the same procedure: integrate from a guess, compare with what is wanted or measured, use the state transition matrix to relate small changes in the state to small changes in the result, solve the linear equations, repeat. For orbit determination there are more measurements than unknowns, and the linear solve becomes a least-squares fit weighted by the measurements’ precision; Gauss invented the method to recover the orbit of Ceres, which had been lost behind the Sun after forty days of observation, and the modern navigation filter is its descendant.

The two run together throughout a mission. Tracking measures the craft’s position and velocity with an uncertainty, and the matrix carries that uncertainty forward to the arrival, where it becomes an ellipse in the aiming plane; the correction burn is chosen to put the ellipse’s centre on the target, and the next weeks of tracking shrink the ellipse until the next burn. An error that is nearly all in one direction — long along the track, narrow across it — is exactly what the along-track growth in the second figure produces, and it is the shape every arrival ellipse has.

What the calculation leaves out #

The integration is two-dimensional and uses circular, coplanar orbits for the Earth, Mars and Jupiter; the real problem is three-dimensional, with eccentric, inclined orbits, and the matrix is six by six. The craft is a point with no thrust except impulsive burns, no radiation pressure, and no gravity from the Earth or Mars — the departure and arrival hyperbolas are omitted entirely, which is why the sensitivities are those of the heliocentric leg alone. The matrix is computed by finite differences, which is simple and adequate here but loses precision for very long or very chaotic trajectories, where it is integrated from the linearised equations instead. And the integration’s precision, a few millimetres at arrival, is set by its step size; the operational integrators that navigate real missions are several orders of magnitude more precise, which is why their corrections can continue to converge where these stop.

Still open: trajectories with no good first guess #

Differential correction converges only if it starts close enough, and the patched conic supplies that start for any trajectory that is mostly a sequence of two-body arcs. It fails for trajectories that are not: the low-energy transfers that drift slowly through the regions where the Sun’s and a planet’s gravity are comparable, following the manifolds that thread the necks between regions of the three-body problem, where no conic is a good approximation and the sensitivity of the arrival to the departure is not seventy-six kilometres per millimetre a second but effectively unbounded. Finding a first guess there needs the dynamical structure of the three-body problem itself, and how to find such guesses systematically — rather than by searching — is one of the live questions of trajectory design. The ordinary interplanetary trajectory is well conditioned enough for Newton’s method; the cheapest ones are the least.