Astronomy & Celestial Mechanics
the sky, computed
Spaceflight

A position measured from a frequency

A spacecraft is unresolvable and unreachable, and everything known about where it is comes from two scalars — a round-trip light time and a Doppler shift. Neither is an angle. The orbit solution returns two angles anyway, because the antenna is bolted to a rotating planet.

Assumes Orbit determination, Ephemerides and Patched conics.

18 min read 9 figures Everything is a conicTwo bodies, and no moreThe sky as a clock

An asteroid is found by measuring where it is. Five directions and no distance among them go into an orbit determination, the angles are the observable, and the distance falls out of the geometry.

A spacecraft is the opposite case in every respect. It is a point of reflected sunlight far too faint to image and far too small to resolve, but it carries a radio transmitter, so what can be measured about it is not where it is but how far away it is and how fast that is changing — a round-trip light time and a Doppler shift, both scalars, neither of them an angle.

The orbit solution nevertheless returns a right ascension and a declination to nanoradians. The reason is that the antenna doing the measuring is attached to a planet that turns.

An eight-hour pass, and a 351 m s⁻¹ sinusoid that is the whole of the angle. Above: the range rate a two-way Doppler measurement returns over one pass from Goldstone, for a spacecraft receding at 14.6 km s⁻¹. Nothing here is an angle. The measurement is the fractional shift of a carrier the spacecraft coherently turned around and sent back, and its interpretation is that the distance is changing at some rate. Below: the same data with the spacecraft's own smooth signature removed. What is left is a sinusoid of exactly one cycle per day — the station's own motion, carried east at 379 metres a second by the rotation of the Earth, projected onto the line of sight. Its amplitude is that speed times cos δ and returns a declination of 22.0°; its zero crossing is the moment the spacecraft passed the meridian and returns the right ascension. The Earth's rotation is the interferometer. With Doppler good to 0.05 mm s⁻¹ at a 60-second cadence, 480 samples fit that amplitude to 0.003 mm s⁻¹ and the declination to 23 nanoradians — which is 4.7 milliarcseconds, from an instrument with no image plane and no angular resolution of any kind. What the picture cannot show is the part that makes this hard in practice: the spacecraft's own signature is not a straight line but a trajectory with unmodelled accelerations in it, and separating a slow non-gravitational force from a slow drift in the angles is the whole art of the fit.
Fig. 1 One tracking pass. Above: the range rate a two-way Doppler measurement returns over eight hours, for a spacecraft receding at fifteen kilometres a second. Nothing in it is an angle. Below: the same data with the spacecraft’s own smooth signature removed, leaving a sinusoid of exactly one cycle per day — the station’s own motion, carried east at 379 metres a second by the rotation of the Earth, projected onto the line of sight. Its amplitude is that speed times cos⁡δ\cos\delta and returns the declination; its zero crossing is the moment of meridian passage and returns the right ascension.

What a two-way Doppler measurement is

The word two-way is load-bearing. A one-way measurement — a transmitter on the spacecraft, a receiver on the ground — would compare the received frequency against a standard on the ground, and its accuracy would be set by the stability of the oscillator on board, which is a small, cold, unmaintained device.

A two-way measurement sends a carrier up from a hydrogen maser on the ground, and the spacecraft’s transponder returns it coherently: it multiplies the received frequency by an exact rational number and retransmits, so the returned signal’s phase is locked to the transmitted one. The spacecraft contributes no frequency of its own. What comes back is compared with what went out, at the same station, against the same maser, and the difference is

Δff  =  − 2ρ˙c,\frac{\Delta f}{f} \;=\; -\,\frac{2\dot\rho}{c},

with ρ˙\dot\rho the rate of change of the one-way distance. A ten-hertz shift on an X-band carrier at eight gigahertz is a range rate of two tenths of a millimetre a second.

The quantity actually recorded is not the instantaneous shift but the cycle count over a fixed interval — the number of carrier cycles received in sixty seconds compared with the number transmitted — which is an integral of the range rate and therefore a change of range over that interval, measured to a small fraction of a wavelength. Modern deep-space Doppler reaches 0.02 to 0.05 millimetres a second at a sixty-second count time.

Where the angles come from

The station’s velocity has three parts: the Earth’s orbital motion about the barycentre, the Earth’s rotation, and the station’s height above the geoid. The first is large, smooth and modelled from an ephemeris. The second is the measurement.

A station at latitude ϕ\phi is carried east at ωRcos⁡ϕ\omega R\cos\phi — 465 metres a second at the equator, 379 at Goldstone’s 35.4 degrees. Projected onto the direction of a source at right ascension α\alpha and declination δ\delta, that gives a contribution to the range rate of

ρ˙stn  =  − ωRcos⁡ϕ cos⁡δ sin⁡H,H=θLST−α,\dot\rho_{\rm stn} \;=\; -\,\omega R\cos\phi\,\cos\delta\,\sin H, \qquad H = \theta_{\rm LST} - \alpha ,

a sinusoid of exactly one cycle per sidereal day whose amplitude is proportional to cos⁡δ\cos\delta and whose phase is α\alpha. Fit the sinusoid and both angles come out.

That is the whole mechanism, and it is worth stating what it means: the Earth’s rotation is the interferometer. A single antenna, sampling its own carousel over eight hours, measures the same thing two antennas separated by a baseline of one Earth radius would.

An eight-hour pass, and a 130 m s⁻¹ sinusoid that is the whole of the angle. Above: the range rate a two-way Doppler measurement returns over one pass from Goldstone, for a spacecraft receding at 14.6 km s⁻¹. Nothing here is an angle. The measurement is the fractional shift of a carrier the spacecraft coherently turned around and sent back, and its interpretation is that the distance is changing at some rate. Below: the same data with the spacecraft's own smooth signature removed. What is left is a sinusoid of exactly one cycle per day — the station's own motion, carried east at 379 metres a second by the rotation of the Earth, projected onto the line of sight. Its amplitude is that speed times cos δ and returns a declination of 70.0°; its zero crossing is the moment the spacecraft passed the meridian and returns the right ascension. The Earth's rotation is the interferometer. With Doppler good to 0.05 mm s⁻¹ at a 60-second cadence, 480 samples fit that amplitude to 0.003 mm s⁻¹ and the declination to 9 nanoradians — which is 1.9 milliarcseconds, from an instrument with no image plane and no angular resolution of any kind. What the picture cannot show is the part that makes this hard in practice: the spacecraft's own signature is not a straight line but a trajectory with unmodelled accelerations in it, and separating a slow non-gravitational force from a slow drift in the angles is the whole art of the fit.
Fig. 2 The same pass for a spacecraft at seventy degrees of declination, where the diurnal sinusoid has shrunk to a third of its equatorial amplitude. That is cos⁡δ\cos\delta doing exactly what the expression says, and it is the measurement rather than a degradation of it: the amplitude is the declination, so a small sinusoid is a large declination and is read off just as cleanly. What changes is the leverage on the other angle. The right ascension comes from the sinusoid’s phase, and a phase is recovered from a small sinusoid worse than from a large one, so a high-declination target is well determined in one coordinate and poorly in the other — the opposite failure to the one the next section is about.

What it is worth, and where it fails

The precision follows from the noise and the number of samples. With Doppler good to 0.05 millimetres a second at a one-minute cadence, an eight-hour pass gives 480 samples, and fitting a sinusoid of known frequency to them determines its amplitude to σ2/N\sigma\sqrt{2/N} — about three micrometres a second.

Dividing by the station velocity gives an angle. But the two angles are recovered from the same sinusoid in different ways, and they fail in different places.

Two angles from one sinusoid, and each of them fails where the other is best. What a pass of 8 hours of 0.05 mm s⁻¹ Doppler from Goldstone is worth as an angle, against the declination of the spacecraft. The diurnal signature carries both angles and carries them differently. Its amplitude is the station's east velocity times cos δ, so inverting it for the declination divides by sin δ and the error runs away at the celestial equator: at 2° it is 244 nanoradians, thirty times worse than at 45°. Its phase is the right ascension directly, and recovering a phase from a sinusoid of amplitude proportional to cos δ divides by that cosine, so the right ascension is the one that fails near the pole. The two curves cross at exactly 45° because 1/sin and 1/cos do. Neither failure is a defect of the instrument, and neither can be reduced by tracking longer: at zero declination the station's motion is entirely along the line of sight all day, so no amount of it contains any information about how far north the spacecraft is. The geometry is degenerate rather than noisy. That is the whole reason a deep-space network also observes quasars: a differenced one-way range against a source of known position measures the plane-of-sky angle directly and does not care about the declination at all — and every Mars mission needs it, because a transfer to Mars spends most of its cruise within ten degrees of the equator.
Fig. 3 The two errors against declination. The amplitude gives cos⁡δ\cos\delta, so inverting for the declination divides by sin⁡δ\sin\delta and the error runs away at the celestial equator — at two degrees it is thirty times worse than at forty-five. The phase gives the right ascension directly, and recovering a phase from a sinusoid whose amplitude is proportional to cos⁡δ\cos\delta divides by that cosine, so the right ascension fails near the pole. The two curves cross at exactly 45°, because 1/sin⁡1/\sin and 1/cos⁡1/\cos do.

The low-declination failure is not noise and cannot be reduced by tracking longer. At δ=0\delta = 0 the station’s motion lies entirely in the plane containing the line of sight, so no amount of it contains information about how far north the spacecraft is. The geometry is degenerate rather than noisy.

An eight-hour pass, and a 378 m s⁻¹ sinusoid that is the whole of the angle. Above: the range rate a two-way Doppler measurement returns over one pass from Goldstone, for a spacecraft receding at 14.6 km s⁻¹. Nothing here is an angle. The measurement is the fractional shift of a carrier the spacecraft coherently turned around and sent back, and its interpretation is that the distance is changing at some rate. Below: the same data with the spacecraft's own smooth signature removed. What is left is a sinusoid of exactly one cycle per day — the station's own motion, carried east at 379 metres a second by the rotation of the Earth, projected onto the line of sight. Its amplitude is that speed times cos δ and returns a declination of 3.0°; its zero crossing is the moment the spacecraft passed the meridian and returns the right ascension. The Earth's rotation is the interferometer. With Doppler good to 0.05 mm s⁻¹ at a 60-second cadence, 480 samples fit that amplitude to 0.003 mm s⁻¹ and the declination to 163 nanoradians — which is 33.5 milliarcseconds, from an instrument with no image plane and no angular resolution of any kind. What the picture cannot show is the part that makes this hard in practice: the spacecraft's own signature is not a straight line but a trajectory with unmodelled accelerations in it, and separating a slow non-gravitational force from a slow drift in the angles is the whole art of the fit.
Fig. 4 Three degrees from the celestial equator, which is where a Mars cruise spends most of its time. The sinusoid is at nearly its full 378 metres a second — the largest signal on offer — and it is the least informative version of it, because the amplitude is cos⁡δ\cos\delta and the cosine is flat at zero. A declination of three degrees and one of six produce sinusoids differing by less than a part in a thousand, so the strongest signature in the data constrains the coordinate that matters least well. That is the shape of a degenerate measurement rather than a weak one: more data, better masers and longer passes all improve the same badly conditioned inversion.

That matters practically because a transfer to Mars spends most of its cruise within ten degrees of the celestial equator — the ecliptic crosses it twice a year and interplanetary trajectories stay near the ecliptic — so the worst case is the normal case, and the launch window that sets the geometry is chosen with no regard to it.

The pancake

Ranging and Doppler measure the same coordinate — distance, and its rate. Both are along the line of sight. The angles come out only through the small diurnal modulation, and they come out as angles, which have to be multiplied by the distance to become a position.

A metre along the line of sight and 4 kilometres across it. Above: the round-trip light time of a ranging signal, against distance. It is a straight line of slope one, which is the point — the delay is the distance, with no model and no calibration between them, and a code correlated to a few nanoseconds gives the radial coordinate of a spacecraft at Mars to about 1 metre. Below: what the same tracking session gives across the line of sight. The angle is good to 50 nanoradians — the Doppler noise alone would give 12, and the difference is the troposphere, the station coordinates and the solar plasma, none of which average down — and an angle times a distance is a length that grows: at Mars's 0.52 au it is 4 kilometres, 3890 times the radial error. The error region of an interplanetary spacecraft is therefore not a sphere but a pancake, thin along the line of sight and enormously wide across it, and every operational consequence follows from that shape: a Mars arrival is aimed at a corridor in the plane of the sky rather than at a distance, the last approach manoeuvres are almost entirely lateral, and the observation that improves the fit most is never another day of Doppler but a single differenced measurement against a quasar. The cheapest observable measures the coordinate that matters least.
Fig. 5 What that does to an error region. Above: the round-trip light time against distance — a straight line of slope one, because the delay is the distance, with no calibration between them; a ranging code correlated to a few nanoseconds fixes the radial coordinate of a spacecraft at Mars to about a metre. Below: what the same session gives across the line of sight. Fifty nanoradians is a superb angle, and an angle times a distance grows: at Mars it is four kilometres, four thousand times the radial error. The error region is a pancake, thin along the line of sight and enormously wide across it.

Every operational consequence follows from that shape.

An arrival is aimed in the plane of the sky. A Mars entry has to hit a corridor about a degree and a half wide in flight-path angle, which at arrival corresponds to a few kilometres of aim point — comparable with the plane-of-sky uncertainty and far larger than the radial one. So the last approach manoeuvres are almost entirely lateral.

The most valuable observation is not more Doppler. Adding a day of tracking improves the radial knowledge, which was already the good part. What improves the bad part is an angular measurement, and there is one: ΔDOR, in which two stations separated by a continental baseline observe the spacecraft and then a quasar of known position a few degrees away, and difference the two. Most of the error sources — station clocks, the troposphere, the ionosphere, the ephemeris — are common to the two observations and cancel. A ΔDOR pass gives the plane-of-sky angle to two to five nanoradians, an order of magnitude better than a day of Doppler, in twenty minutes.

A nanosecond across the Earth is 35.69 nanoradians on the sky. The angular accuracy of a differenced-delay measurement against the length of the baseline it is measured on, both axes logarithmic, for three levels of delay precision. The relation is σ_θ = cσ_τ/B and nothing else, so every curve is a straight line of slope −1.00: the only two ways to measure an angle better are a better clock or a wider Earth, and only one of those is available. The three marked baselines are the ones that exist — the deep-space complexes in California, Spain and Australia, 8,400, 10,600, 11,700 kilometres apart. On the longest of them a delay good to 0.05 nanoseconds is 1.28 nanoradians, which at 0.52 astronomical units is 100 metres across the line of sight; a more typical 0.15-nanosecond measurement on the shortest baseline is 5.35 nanoradians. What makes any of this survivable is that the same pair of antennas observes a quasar a few degrees away immediately afterwards. The quasar is at infinity, its position is known better than the measurement, and subtracting its delay from the spacecraft's removes the clock offsets, the water vapour over each dish and the station coordinates in one step — so the number that comes out is not a delay at all but an angular separation from a fixed point in the sky.
Fig. 6 What a continental baseline buys, drawn as the conversion it is. A nanosecond of differential delay across a baseline the width of the Earth is thirty-six nanoradians on the sky, so timing the same wavefront at two stations to a few tens of picoseconds gives an angle of a few nanoradians directly — as an angle, from a delay, with no diurnal fitting and no dependence on declination. The quasar is what makes the picoseconds achievable: it is at the same place in the sky and at effectively infinite distance, so switching between the two sources every few minutes differences away the clocks, the troposphere and the station coordinates, and what survives is the angle between a spacecraft and a fixed point.

What is really being fitted

The estimator is not solving for a position. It is solving for a trajectory and for the things that corrupt the measurement, simultaneously, over weeks of data.

The state vector contains six orbital elements, but also: a solar radiation pressure coefficient, because sunlight pushes on the spacecraft with an acceleration of order 10−710^{-7} m s⁻²; the magnitude and direction of any thruster firing, including the small ones used to unload momentum wheels; a troposphere delay per station per pass; the station coordinates themselves, which move by centimetres with tides and plate motion; and a bias per pass to absorb whatever is left.

Each of those competes with the signal. A small unmodelled acceleration along the line of sight is indistinguishable, over a few days, from a slightly different initial velocity — and separating them is exactly the same problem as separating a slow drift in the angles from a slow force.

Two interiors, 2.59 mm/s apart, against a floor of 0.02. What a radio link measures when a spacecraft flies past a moon. The horizontal axis is time from closest approach in minutes and the vertical axis is the accumulated change in the line-of-sight velocity in millimetres per second, after the pull of the moon as a point mass has been fitted and removed. What is left is the part of the field that is not spherically symmetric, and the two curves are what two interiors predict for it. The upper one is a body whose tidal Love number is 0.616 — a shell floating on a global liquid layer, free to deform almost as a fluid would. The lower one is a body solid throughout, at 0.03. They differ by 2.59 millimetres per second in the accumulated deflection, against a floor of 0.02 for a coherent two-way X-band link integrated over tens of seconds: 130 standard deviations in a single pass. That ratio is the whole reason the measurement is possible, and it is worth stating what is being compared. The point-mass deflection itself is 837 metres per second — five orders of magnitude larger than the signature of interest — so the interior is not read off the Doppler curve but off the residual left after a model of everything larger has been subtracted: the moon's mass, the planet's, the spacecraft's own thrusting and outgassing, the plasma along the path, the station's motion, and relativity. Every one of those has to be right to a part in a hundred thousand before the last curve here means anything, which is why gravity science needs many passes and a global fit rather than one flyby and a subtraction. The published uncertainty on Titan's k₂ is eleven per cent rather than the fraction of a per cent this signal-to-noise would suggest, and the difference is entirely correlations with the other parameters in that fit — a reminder that a formal error on a curve is not the error on the number extracted from it. The straight-line path assumed here is exact only in the limit of a fast flyby; a slow one bends, and the bending is solved for rather than approximated.
Fig. 7 The same instrument used as a measurement rather than as a navigation aid, which is what the state vector’s competing terms look like when one of them is the point. A spacecraft in orbit about a body responds to that body’s mass distribution, and the Doppler residual between two candidate interiors is a couple of millimetres a second against a noise floor of two hundredths — a hundred to one. That ratio is why the same tracking data that place a spacecraft also weigh the planet it is at: the accelerations being estimated as nuisance parameters during cruise become the signal in orbit, and nothing about the hardware changes.

The delay that is not a measurement

The round-trip light time is the observable, and the same quantity is an operational constraint that shapes how a mission is flown. It is worth separating the two uses, because the second is what makes deep-space navigation a different discipline from tracking a satellite.

At Mars the one-way light time runs from four minutes to twenty-two, depending on where the two planets are. At Saturn it is over an hour. Whatever the spacecraft is doing, the ground learns about it that long afterwards, and any instruction sent arrives that long after it was composed.

For cruise navigation that is a nuisance rather than a barrier: trajectory corrections are planned days in advance and executed on a stored sequence, so a twenty-minute delay is irrelevant. For anything that happens fast it is decisive. An atmospheric entry lasts about seven minutes and the light time to Mars is longer than that, so the entire descent has completed — successfully or otherwise — before the first telemetry from its beginning arrives. Nothing can be commanded, monitored or aborted; the vehicle flies the sequence it was given and the ground watches a recording.

The consequence is that everything which must respond faster than the light time has to be decided on board. Entry guidance, landing hazard avoidance, and the final approach of a rendezvous are all autonomous by necessity rather than by preference, and the navigation solution’s job is to hand the vehicle a state vector accurate enough that its own guidance can take over.

That places the whole burden on the last update. A state vector delivered hours before entry is the last external information the vehicle receives, and the pancake-shaped error region described above is what it carries into the atmosphere. The light time converts a navigation accuracy into a design requirement: how autonomous the vehicle must be is set by how long the ground takes to notice anything, which is set by the speed of light and the geometry of two orbits.

The generalisation

Two ideas here are more general than the application, and both are about what a measurement’s geometry buys.

A moving observer measures angles. Any observable that is a projection of a known velocity onto an unknown direction carries that direction in the phase and amplitude of its variation. The same argument gives a stellar parallax from the Earth’s orbit, an aberration ellipse from the Earth’s velocity, a pulsar’s position from the annual modulation of its arrival times, and a spacecraft’s declination from a station’s carousel. In every case the baseline is the observer’s own motion, and the precision is the measurement noise divided by that motion — so a bigger orbit or a faster rotation is worth exactly as much as a better instrument.

An error ellipsoid is a shape, not a size. Quoting a navigation accuracy as one number is nearly always wrong, because the good and bad directions differ by orders of magnitude and the mission cares about a particular one. The same is true of an asteroid’s orbit, whose error is nearly all along-track, of a lensing mass measurement, and of any fit whose parameters are correlated. The question is never how well something is known but how well the combination that matters is known, and the answer usually depends on when the next observation is taken rather than on how good it is.

How a range is actually measured

The light time was described above as though a pulse were sent and its echo timed. It is not, and the way it is really done introduces an ambiguity worth knowing about.

A pulse bright enough to time against the noise would need far more power than a spacecraft’s transponder returns. What is transmitted instead is a continuous phase modulation of the carrier by a long pseudorandom code — a sequence of ones and zeros with the statistical properties of noise and a known, reproducible pattern. The returned signal is correlated against a delayed copy of the same code, and the delay that maximises the correlation is the round-trip time.

The advantage is that the correlation concentrates the whole integration’s energy into one peak: a signal far below the noise in any instant is recovered by correlating over minutes. The precision is set by the code’s chip rate, and a few nanoseconds of timing corresponds to a metre of range.

The ambiguity is that the code repeats. A correlation peak is found at the delay modulo the code’s repetition period, so the measurement gives the range modulo a fixed distance — for a code repeating every second, modulo 150,000 kilometres. Which repetition is the right one has to come from somewhere else, and it comes from the orbit solution: the predicted range is known to far better than the ambiguity interval, so the integer is unambiguous.

That is a mild circularity and it is a real one. A ranging measurement is only interpretable given an orbit good enough to resolve the ambiguity, which is why the first range after a long gap in tracking, or after an unplanned manoeuvre, is the one that has to be treated carefully. In practice the codes are built as products of components of different lengths, so the ambiguity can be resolved in stages from a coarse prior, and the resolution is checked rather than assumed.

A metre along the line of sight and 4 kilometres across it. Above: the round-trip light time of a ranging signal, against distance. It is a straight line of slope one, which is the point — the delay is the distance, with no model and no calibration between them, and a code correlated to a few nanoseconds gives the radial coordinate of a spacecraft at Mars to about 3 metre. Below: what the same tracking session gives across the line of sight. The angle is good to 50 nanoradians — the Doppler noise alone would give 12, and the difference is the troposphere, the station coordinates and the solar plasma, none of which average down — and an angle times a distance is a length that grows: at Mars's 0.52 au it is 4 kilometres, 1297 times the radial error. The error region of an interplanetary spacecraft is therefore not a sphere but a pancake, thin along the line of sight and enormously wide across it, and every operational consequence follows from that shape: a Mars arrival is aimed at a corridor in the plane of the sky rather than at a distance, the last approach manoeuvres are almost entirely lateral, and the observation that improves the fit most is never another day of Doppler but a single differenced measurement against a quasar. The cheapest observable measures the coordinate that matters least.
Fig. 8 The pancake drawn with the radial error three times larger — a coarser ranging code, or a session degraded by the corona — and it is still a pancake. Three metres against four kilometres is a ratio of thirteen hundred rather than four thousand, which is the same picture with slightly less contrast. That insensitivity is the argument for spending the effort on angles rather than on ranging: the radial coordinate is so far ahead of the transverse ones that degrading it by a factor of three changes nothing about which measurement the navigation is short of. It is also why the ambiguity resolution described above is never the limiting step — the prior orbit is known transversely to kilometres and radially to metres, and the ambiguity interval is a hundred and fifty thousand kilometres.

The Doppler measurement has an ambiguity of its own, of a different kind and with the same character. What is counted is a number of cycles over an interval, so what is recovered is a change in range rather than a range — the constant of integration is not in the data. That constant is supplied by the ranging measurement, which is why the two data types are always taken together even though Doppler alone is far more precise for a rate.

The pairing is a good illustration of how the observables complement each other. Ranging fixes the absolute distance badly and unambiguously; Doppler fixes its rate superbly and knows nothing about its value. Combining them gives a distance that is accurate at every epoch and drifts only as slowly as the Doppler noise allows, which is far better than either supplies alone.

A measurement with an ambiguity is a measurement plus an integer, and where the integer comes from a model, the model is part of the observation.

Where the model stops

The solar plasma. Radio waves passing near the Sun are delayed and dispersed by the corona, and the delay varies as the plasma moves. Near solar conjunction the effect swamps everything, tracking data are downweighted or discarded, and a mission’s navigation quietly degrades for weeks. The dispersion is a function of frequency, so observing at two bands measures and removes it, which is one of the reasons deep-space links carry both X and Ka band.

The troposphere. The wet component of the atmospheric delay varies by centimetres on timescales of minutes and is the dominant error in ΔDOR. It is estimated from water-vapour radiometers pointed along the same line of sight, which works and is not perfect.

And the spacecraft itself. Every unmodelled acceleration enters the fit as a trajectory error. The most famous case took thirty years to close: Pioneer 10 and 11 showed a small anomalous deceleration towards the Sun, of order 8×10−108\times10^{-10} m s⁻², which survived every attempt to attribute it to a modelling error until a thermal analysis of the spacecraft’s own radiated heat — the recoil from the anisotropic re-radiation of a decaying plutonium source — accounted for it. The same physics as an asteroid’s orbit moved by heat, on a spacecraft, mistaken for new gravity for three decades.

What was actually flown

Numbers, from missions whose navigation is documented.

Cassini’s arrival at Saturn in 2004 required threading a gap between the F and G rings some 25,000 kilometres wide, after a seven-year cruise. The delivered plane-of-sky error at arrival was of order ten kilometres, achieved with a tracking campaign that combined continuous Doppler, periodic ranging and ΔDOR passes every few days in the final months.

Mars Reconnaissance Orbiter’s entry into orbit in 2006 needed the periapsis of the arrival hyperbola placed within a few kilometres of a target 400 kilometres above the surface. The navigation error at the final trajectory correction was under a kilometre in the plane of the sky and under a hundred metres radially — the pancake, at working scale.

And the counter-example. Mars Climate Orbiter was lost in 1999 because the small forces produced by momentum-wheel desaturations were supplied to the navigation team in pound-force-seconds and used as newton-seconds. The error was a factor of 4.45 in an acceleration the fit was estimating anyway, and it accumulated over nine months into a periapsis some 80 kilometres lower than intended. Nothing about the radiometric data was wrong; the failure was in one of the models the data are interpreted through, which is where a navigation failure nearly always is.

One more baseline shows what the interferometric technique buys over the Doppler one.

A nanosecond across the Earth is 35.69 nanoradians on the sky. The angular accuracy of a differenced-delay measurement against the length of the baseline it is measured on, both axes logarithmic, for three levels of delay precision. The relation is σ_θ = cσ_τ/B and nothing else, so every curve is a straight line of slope −1.00: the only two ways to measure an angle better are a better clock or a wider Earth, and only one of those is available. The three marked baselines are the ones that exist — the deep-space complexes in California, Spain and Australia, 8,400, 10,600, 11,700 kilometres apart. On the longest of them a delay good to 0.05 nanoseconds is 1.28 nanoradians, which at 0.52 astronomical units is 100 metres across the line of sight; a more typical 0.15-nanosecond measurement on the shortest baseline is 5.35 nanoradians. What makes any of this survivable is that the same pair of antennas observes a quasar a few degrees away immediately afterwards. The quasar is at infinity, its position is known better than the measurement, and subtracting its delay from the spacecraft's removes the clock offsets, the water vapour over each dish and the station coordinates in one step — so the number that comes out is not a delay at all but an angular separation from a fixed point in the sky.
Fig. 9 The differenced-ranging geometry with a two-nanoradian systematic floor. A nanosecond of differential delay across the Earth is about thirty-six nanoradians on the sky, so the technique measures a spacecraft’s position to a few nanoradians — a quantity the Doppler shift alone contains almost none of.

Where this ladder goes next

Later rungs on this anchor: ΔDOR in detail, and why differencing against a quasar removes almost every systematic at once; optical navigation, where an onboard camera images the target against background stars and supplies exactly the plane-of-sky information the radio link cannot; onboard autonomous navigation, which is forced by the light time as soon as the round trip exceeds the time available to react; the use of tracking data as science, since the same Doppler that navigates a spacecraft measures a planet’s gravity field when the spacecraft is in orbit around it; and relativistic light-time corrections, which are not a refinement but a several-hundred-metre effect that every solution has carried since the 1960s.

Where this goes next

Ceres from five directions and no distance. Ceres seen five times over 41 days, from an Earth on a circular orbit, reduced in the plane. Each sighting gives a direction and no range, so the object is somewhere on its sight line; the five lines here span 1.37° of geocentric arc altogether, and Earth's own motion supplies the only baseline there is — 0.403 AU of its 0.691 AU of travel lies across the sight lines. A planar orbit is four numbers, so five angles over-determine it and one orbit comes out: a = 2.7658 AU, e = 0.0785. That is the answer and not an input — the sightings were generated at a = 2.7658 AU and e = 0.0785, and the solve, which sees only the directions and the dates, returns them to 3e-12. The two shaded sectors are what closes the determination: between the first and middle sightings the radius vector sweeps 0.2993 AU² in 21.0 days and between the middle and last 0.2853 AU² in 20.0 days, a ratio of 1.04918 against a time ratio of 1.04918. Slide all three crossings out along their sight lines together and that equality fails at once, so it fixes the distance by itself, with no propagation anywhere in the argument — and it gives a = 2.7658 AU over again. The two dashed curves are candidates that thread the same three sight lines at 85% and 108% of the recovered distance: a = 1.86 AU at e = 0.35, sweeping its areas in a ratio 4.5% wrong; and a = 5.61 AU at e = 0.49, sweeping its areas in a ratio 2.9% wrong. A few per cent in the distance is an orbit of another kind, which is the same fact the conditioning panel measures: one arcsecond of angle error moves a by 0.60% on this arc. Five directions and no distance among them An image of a moving point records an angle and throws the range away, so an orbit has to be assembled out of angles alone. How many angles are needed is not a detail of the method — it is the whole of what a determination is. After one revolution the error is 3π times longer than it is wide, and after 300 it is 2827. The two semi-axes of a fitted orbit's position uncertainty, against elapsed revolutions, for a solution whose semi-major axis is uncertain by 12 kilometres. The radial extent does not grow at all: a body on a slightly larger orbit is slightly further out and stays so. The along-track extent grows linearly, because δn/n = −(3/2)δa/a makes a semi-major-axis error into a mean-motion error and a mean-motion error into a phase that runs away — a·δM = 3πN·δa after N revolutions. The ratio is 3π ≈ 9.42 after a single revolution and 2827 after 300, which is why an asteroid recovered after one apparition is found within a few arcseconds of its predicted place along its own track and could be a long way from it in time. Every consequence of this in practice — that an impact probability is a one-dimensional integral rather than a volume, that a keyhole is an interval, that the next observation worth taking is the one across the track rather than the one that fits best — is a restatement of these two lines diverging. An error that is nearly all in one direction A fitted orbit's uncertainty is not a ball. Within a few revolutions it has collapsed onto a line along the track, because an error in the size of an orbit is an error in its period and an error in period is a phase that runs away — which is why an impact probability is an integral along a curve rather than over a volume. Mars to five metres and Neptune to five thousand kilometres, in the same file. Present-day heliocentric position uncertainty for each planet, in kilometres, with the range component marked separately below it. The two differ because a transponder measures a distance along the line of sight and says nothing about the two directions across it, so a planet with an orbiter is known radially some 17 times better than it is known altogether. Neptune is 10⁶ times less well determined than Mars and only 20 times further away, which is the whole point: the accuracy is a property of the observations, not of the geometry. Mars has carried a transponder almost continuously since 1976; Neptune has been visited once, in 1989, and everything else known about it is meridian-circle astrometry covering 1.07 of one orbit. The two ice giants are the only entries here whose ephemerides are still limited by nineteenth-century technology, and the only cure is a spacecraft. The table that is a fit A planetary ephemeris is not evaluated from Kepler's laws and is not evaluated from a theory. It is a numerical integration whose starting conditions were least-squares fitted to a century and a half of observations, and its accuracy is a property of those observations rather than of the mathematics.

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Coherent transponderDeclinationDeep-space networkDelta-DORDiurnal signatureLight timeNavigationOrbit determinationPlane of skyRangingSystematic errorTwo-way Doppler