Four media between the antenna and the spacecraft

A deep-space navigation measurement is a time. A signal leaves a station on Earth, is transponded by the spacecraft, and returns; the round-trip time is measured to a nanosecond, which is fifteen centimetres of range at a distance of hundreds of millions of kilometres.

That precision is spent almost entirely on things that are not the trajectory. The signal passes through four media on each leg, and each of them delays it by more than the measurement’s precision — some by seven orders of magnitude more.

A position measured from a frequency is the technique; this essay is about the four things standing between the frequency and the position, and about the one property that lets two of them be removed exactly rather than modelled.

The four, and what each depends on #

The troposphere. The neutral atmosphere delays a radio signal by about 2.3 metres at the zenith, of which 2.0 is the dry component — proportional to the surface pressure and calculable to millimetres — and 0.3 is water vapour, which is neither steady nor predictable. The delay scales with the path length through the atmosphere, so it rises roughly as the cosecant of the elevation. It has no frequency dependence at radio wavelengths.

The ionosphere. Free electrons in the upper atmosphere delay the group velocity by an amount proportional to the total electron content along the path and to the inverse square of the frequency. At X band it is centimetres by day and millimetres by night; at S band, nine times more.

The solar plasma. The same physics as the ionosphere, in the solar wind, and with a total electron content that rises steeply as the line of sight approaches the Sun. At a few degrees of elongation it dominates everything; at a right angle it is comparable to the ionosphere.

The Sun’s gravity. The Shapiro delay, which is not a medium at all: it is the extra light-travel time through the curved region near the Sun. It rises as the logarithm of the impact parameter and it has no frequency dependence whatsoever.

A fifth term is sometimes listed and is not a medium either: the station’s own position, which moves by centimetres with the solid Earth tide and by decimetres with plate motion. It is modelled rather than measured, and it is included here only to mark the boundary — everything beyond the antenna’s phase centre is the path, and everything behind it is the station.

That last property is the key to the entire calibration strategy.

It is worth putting the four in order of size at a typical geometry, because the ordering is not what intuition suggests. Away from the Sun the troposphere is the largest by far — several metres — and it is also the best calibrated, because it is measured directly rather than modelled. The Shapiro delay is next at tens of metres near conjunction and a few metres away from it, and it is computed exactly. The plasma terms are the smallest away from conjunction and the largest near it, and they are the only ones that fluctuate on the timescale of a measurement. So the largest term is not the limiting one, and the limiting one is the one that varies.

Why two frequencies separate two of them #

Two of the four delays scale as the inverse square of the frequency and two do not. Transmitting and receiving at two well-separated frequencies therefore measures the plasma terms directly: the difference between the two ranges is proportional to the electron content and to the difference of the inverse squares, and the difference cancels everything with no frequency dependence.

That gives a plasma calibration measured rather than modelled, valid instantaneously, and accurate to the level the two frequencies can be compared. It is the reason modern deep-space links are dual-frequency, and it is why the Cassini measurement of the relativistic delay coefficient was two orders of magnitude better than its predecessors: not because the delay was measured better, but because the plasma that swamps it was removed rather than modelled.

What remains after the plasma is removed is the troposphere, which is handled by a water-vapour radiometer pointed along the line of sight, and the Shapiro delay, which is computed from theory. The first is a measurement with an accuracy of a few millimetres; the second is a calculation whose accuracy is the accuracy of the relativistic parameter, which is a part in $10^5$.

There is an important asymmetry between the two plasma terms that the shared frequency dependence conceals. The ionosphere is above the station and is therefore common to everything observed from that station at that moment, so it can be measured independently — by the same navigation satellites everybody else uses, whose dual-frequency signals map the electron content over the site continuously. The solar plasma is along the specific line of sight to the spacecraft and has no independent measurement at all. So one of the two is calibrated by an external network and the other only by the spacecraft’s own link, and when the link is single-frequency the second is a model with no check on it.

A delay and its rate are two different problems #

Everything above is stated as a delay, and half the navigation data is not a delay. Doppler measures the rate of change of the path length, so a medium enters it through its time derivative — and the ordering of the four terms is completely different in that currency.

A constant delay is invisible to Doppler. The troposphere’s 2.3 metres at the zenith contributes nothing at all to a range-rate measurement while the elevation is unchanging; what contributes is the change as the station rotates, which over a pass is several metres and is therefore a signal of a millimetre a second or so against a noise of a few hundredths. The dry component’s change is calculable from the elevation to a fraction of a per cent. The wet component’s is not, because water vapour moves.

So the four media are ranked one way for range and another way for Doppler:

• For range, the largest term is the troposphere away from conjunction and the solar plasma near it, and the accuracy is limited by whichever is worst calibrated.
• For Doppler, the largest term is whichever varies fastest over a pass — the wet troposphere at low elevation, and the solar plasma always, because coronal structure crosses the line of sight on timescales of minutes.

The distinction matters because it decides what a medium error looks like in the orbit fit. A slowly varying delay error maps into the spacecraft’s range and is largely absorbed by the along-track position, which is the component a trajectory is least sensitive to anyway. A rapidly varying one maps into the velocity, and a velocity error at a planetary encounter is what the whole measurement chain exists to prevent.

The Shapiro term is the cleanest case of the distinction. Its magnitude near conjunction is tens of metres, which is enormous in range, and its rate of change is smooth and slow, which makes it almost harmless in Doppler except through the very geometry it is being used to measure. That is why the relativistic coefficient is extracted from the shape of the delay’s variation through conjunction rather than from its size at any one moment — the size is degenerate with the trajectory, and the shape is not.

There is a further wrinkle worth having, because it is a genuine sign error waiting to happen. Plasma retards the group velocity and advances the phase velocity, so a plasma disturbance lengthens the measured range and shortens the measured phase path at the same instant. A pipeline that applies a plasma correction with one sign to both data types corrupts one of them by twice the correction, and the resulting inconsistency between range and Doppler is one of the standard diagnostics that something in the media calibration has gone wrong.

What the conjunction does #

The steep rise of the plasma term towards small elongations has two consequences that pull in opposite directions.

Navigation stops. As a spacecraft passes behind the Sun the plasma delay rises by orders of magnitude and fluctuates on timescales of minutes as blobs of corona cross the line of sight. Range residuals of hundreds of metres are normal, and the orbit determination degrades to the point where the tracking is not used. Missions plan around solar conjunctions the way a sailing ship plans around a season.

And the physics becomes measurable. The same geometry that ruins the navigation is where the Shapiro delay is largest, so a conjunction is the one opportunity to measure the relativistic coefficient. The two effects are separated by their frequency dependence, which is why the measurement requires exactly the calibration technique the navigation requires.

There is a third consequence and it is operational rather than scientific. Because the plasma delay is a group delay and the Doppler is a phase rate, the two are affected with opposite signs — the group velocity is reduced and the phase velocity increased. So a plasma disturbance produces a range error and a range-rate error that are anticorrelated, which is a signature an orbit fit can be taught to recognise. Several navigation systems exploit that: the combination of range and Doppler that is insensitive to plasma is formed explicitly, at the cost of some precision, when the geometry demands it.

What was actually measured #

Three measurements establish the sizes and one establishes the method.

The tropospheric zenith delay, from water-vapour radiometry. Pointing a microwave radiometer along the line of sight measures the water vapour’s emission and therefore its delay, to about three millimetres. That is better than any model, it is available in real time, and it is the reason the wet troposphere has stopped being the limiting term.

The plasma, from dual-frequency differencing. The measured total electron content along the line of sight agrees with independent measurements from ionospheric sounders and from solar-wind models, and its removal reduces the range residual scatter by an order of magnitude near conjunction.

The Shapiro coefficient, from Cassini. A measurement of the relativistic parameter to $2\times10^{-5}$, made at solar conjunction with a multi-frequency link. It is the tightest constraint on that parameter from any experiment, and it was possible only because the plasma — larger than the signal at that geometry — was measured at a second frequency rather than modelled.

And the residual after everything. A modern deep-space range residual is a few metres over a pass. Every term in this essay is between one and a thousand times that before calibration, and the fact that the residual is what it is means that all four have been removed to about a part in a thousand of themselves.

Where the picture stops #

Three, and the third is the one that generalises.

The dual-frequency correction has its own noise. Differencing two ranges to remove the plasma amplifies the thermal noise of both, by a factor set by the frequency ratio. For a weak link the calibrated range is noisier than the uncalibrated one, and there is a signal-to-noise below which modelling is better than measuring.

Water vapour is not the only troposphere. The dry component is calculable from the surface pressure and the wet from radiometry, and there is a residual from horizontal gradients — the atmosphere above a station is not horizontally uniform, and the mapping function that converts a zenith delay to a slant delay assumes it is. At low elevations that is the limiting term.

And the calibration is a model of the medium, not of the measurement. Every correction above is applied to the data before the orbit is fitted, which means an error in a correction is absorbed into the orbit. The residuals do not reveal it, because the fit has adjusted to it. That is the structural hazard of any calibration applied upstream of a fit, and it is why the corrections are cross-checked between data types — range, Doppler and angular measurements carry the media differently, and a media error shows up as a disagreement between them rather than as a large residual in any one.

There is also a fourth that deserves stating because it is where the discipline is heading. Optical communication — a laser link rather than a radio one — has no plasma delay at all, because the plasma’s effect scales as the inverse square of the frequency and an optical frequency is five orders of magnitude higher. It has a tropospheric delay that is different in character, it cannot operate through cloud, and it offers ranging precision orders of magnitude better than radio. The media calibration problem for an optical link is a completely different problem with a different largest term, and the fact that its largest term is weather rather than plasma changes where the effort goes.

Why the frequency dependence is the whole design #

The general point is worth pulling out because it is the same one that recurs whenever a contaminant has to be removed.

A contaminant with a known dependence on an observable parameter is not a contaminant — it is a second observable. The plasma delay depends on frequency and the trajectory does not, so observing at two frequencies measures the plasma. That is not a clever trick; it is the only structural way to separate two effects that arrive in the same number, and it appears everywhere in this collection: dust separated from distance by observing in two colours, a pulsar’s dispersion measure separated from its distance by observing at two radio frequencies, an atmospheric prism separated from a source’s spectrum by its dependence on airmass.

The corollary is a design principle rather than an analysis technique. If a measurement will be limited by a contaminant, the question to ask before building the instrument is what the contaminant depends on that the signal does not — and then to arrange to observe along that axis. A dual-frequency link costs a second transmitter, a second receiver and a second antenna feed, and it buys a term that no amount of modelling could have removed.

One further observation about the epoch this all belongs to. Before dual-frequency links, missions near conjunction simply stopped navigating and coasted; the plasma was modelled with a solar-wind model whose accuracy was a factor of two, and the resulting range errors were kilometres. The change from modelling to measuring the plasma happened over about fifteen years and it is the single largest improvement in deep-space navigation accuracy in the discipline’s history — larger than any improvement in clocks, antennas or receivers. Notice that it came from adding an observable rather than from improving one, which is the pattern this collection keeps meeting: a second measurement that depends differently on the contaminant beats a better version of the first.

It is also worth noticing what the four media have in common, since they are otherwise unrelated: all four are along the path rather than at either end. A trajectory model describes where the spacecraft is; a station model describes where the antenna is; and everything in between is the subject of this essay. That three-way division — source, path, receiver — is the standard anatomy of any propagation measurement, and in this case the path term is the largest and the least stable of the three. The same anatomy applies to a planet’s observed position, where the path contributes the light-time and the deflection, and to a pulsar’s arrival time, where it contributes the dispersion.

Where the ladder goes next #

The obvious next rung is the tropospheric mapping function: how a zenith delay is converted to a slant delay, why horizontal gradients break it, and what the residual costs at low elevation. Further up sits the use to which all this precision is put — an ocean inferred from a Doppler residual, where the science is what is left after every term in this essay has been removed.