An ocean found in a Doppler residual

A spacecraft’s position is measured from a frequency: a signal is sent up, transponded coherently, received back, and the round-trip Doppler shift gives the line-of-sight velocity to a fraction of a millimetre per second. That precision is what makes deep-space navigation possible.

It also makes something else possible, which was not the original purpose. A spacecraft is a test particle, and a test particle’s trajectory records the field it moved through. If everything else about that trajectory can be modelled, the residual is the field — and the field, for a body nobody can enter, is the only access to the inside.

What the link actually does #

A one-way Doppler measurement would require the spacecraft to carry a clock as good as the ground station’s, which no spacecraft does. So the measurement is two-way: the ground transmits a carrier locked to a hydrogen maser, the spacecraft’s transponder multiplies it by a fixed rational number and sends it straight back, and the ground compares what returns with what it sent.

The comparison is of phase rather than of frequency, accumulated over an integration time of tens of seconds. Counting cycles of an X-band carrier at eight gigahertz over sixty seconds is counting about half a trillion cycles, and a fractional error of $10^{-13}$ corresponds to a line-of-sight velocity error of about three hundredths of a millimetre per second.

That is the floor, and it is set by the maser, by the troposphere’s wet delay, and by the plasma along the path. Every one of those is a term that has to be calibrated rather than assumed.

There is a second observable in the same signal that is worth distinguishing, because the two are often lumped together as “tracking”. Ranging measures the round-trip time of a modulation imposed on the carrier, and gives a distance to a metre or two. Doppler measures the rate of change of that distance and gives a velocity to hundredths of a millimetre per second. For a gravity measurement the second is far more useful, because what a field does to a trajectory is an acceleration, and integrating an acceleration once gives a velocity while integrating it twice gives a position that has drifted away from any initial condition.

So gravity science is done in the velocity domain. The observable is a curve of line-of-sight velocity against time through the encounter, and the interior appears as a particular shape in it.

Why the tidal part is the variable part #

A moon’s gravity field, measured on a flyby, is dominated by its mass. Fitting that leaves a residual containing the quadrupole — the departure from sphericity — and a moon’s quadrupole has two parts.

One is static: the permanent bulge produced by its rotation and by the average tide from its primary. That part is indistinguishable from any other contribution to $J_2$, and it says very little about the interior on its own, because a rigid body and a fluid one can have the same permanent figure if the fluid one froze in an earlier shape.

The other part varies. A moon on an eccentric orbit experiences a tide whose strength changes through the orbit, and a deformable body responds by changing shape — so its quadrupole oscillates at the orbital period, with an amplitude proportional to the tidal Love number. To first order in the eccentricity the variable part is $3e$ times the static one, which for Titan’s $e = 0.0288$ is about nine per cent.

That is the signal. It is small, and it is the only part of the field that separates a body that deforms from one that does not.

The consequence for mission design is stark: a single flyby cannot do it. What is needed is several flybys at different points of the moon’s orbit, so that the varying part can be seen to vary, and the static part solved for simultaneously. Titan’s Love number came from six such encounters spread over years. It is worth attaching the numbers to that hierarchy, because the ratios are what make the measurement difficult rather than the absolute values. For a thousand-kilometre flyby of Titan the point-mass deflection is around eight hundred metres per second. The static quadrupole contributes something like a metre per second. The variable tidal part is a few millimetres per second. Each step down is a factor of a thousand, and each one has to be modelled to better than a part in a thousand before the next is visible.

That is the shape of every measurement in this essay and it is the reason gravity science is a matter of fitting rather than of reading. Nothing is measured directly; everything is a parameter in a joint solution, and the interior is whatever is left when the rest of the universe has been accounted for.

What has to be removed first #

The chain from a frequency to a Love number passes through a great many models, and it is worth listing them because the final error bar is dominated by them rather than by the noise.

The station. Its position in a terrestrial reference frame, the Earth’s orientation including polar motion and the length-of-day variation, and solid-Earth and ocean tides at the station.

The media. The troposphere’s dry and wet delays, calibrated with water-vapour radiometers; the ionosphere’s dispersion, removed by observing at two frequencies; and the solar plasma along the path, which is the worst of the three near solar conjunction and is the reason gravity-science passes are scheduled away from it.

The planetary ephemeris. The positions of the Sun, the target planet and the moon, each of which has its own uncertainty.

Relativity. The Shapiro delay, the gravitational frequency shift, and the aberration of the light path — all of which are larger than the signal.

The spacecraft itself. Solar radiation pressure, thermal re-radiation from the spacecraft’s own surfaces, anisotropic emission from its radioisotope generators, and any propulsive event including attitude-control thruster firings. This is usually the limiting term, and it is why gravity-science passes are flown with the reaction wheels doing the pointing.

Why the published error is not the noise #

The figure at the top of this essay says a hundred and thirty sigma. The published uncertainty on Titan’s Love number is eleven per cent, which is nine sigma.

The difference is entirely correlations. In a real analysis the Love number is one parameter among dozens solved for simultaneously: the mass, the static $J_2$ and $C_{22}$, the moon’s ephemeris, the spacecraft’s state at each encounter, the non-gravitational accelerations. Several of those produce signatures in the Doppler residual that resemble the tidal one, and the fit cannot fully separate them.

The formal error on a parameter in a joint fit is the square root of the corresponding diagonal element of the inverse of the normal matrix, and when parameters are correlated that is much larger than the error the parameter would have alone. Quoting the single-parameter precision would be reporting the wrong number by an order of magnitude.

This is a general feature of the technique and not a criticism of it. It is also the reason that adding a flyby helps far more than adding integration time to an existing one: a new encounter at a different orbital phase breaks a correlation, and lowering the noise on an existing one does not.

There is a useful diagnostic that follows from all this and that a reader can apply to any published gravity result. Ask how many independent encounters went into it, and at what spread of orbital phase. A field determined from one flyby has its parameters completely entangled; three encounters at similar phases add little; six spread through an orbit are what separate a static quadrupole from a tidal one. The number of flybys is a better guide to the reliability of a result than the number of significant figures.

The signal the whole result rests on scales with two things a mission designer chooses, and both are worth reading at values a real trajectory might have taken.

Where else the same measurement has been made #

Enceladus. Three close flybys measured a gravity field whose $J_2$ and $C_{22}$ ratio departs from the hydrostatic value in a way consistent with a regional or global sea. The gravity result on its own was ambiguous; combined with the libration amplitude it is not.

Europa. Galileo’s flybys gave $J_2$ and $C_{22}$ and hence a moment of inertia of 0.346 — a differentiated body with a metallic core — but not a Love number, because the flybys were too few and not spread in orbital phase. Europa’s tidal response is the primary objective of two missions now on their way.

Jupiter. A close polar orbit measured the zonal harmonics to degree ten and beyond, and read a dilute core out of them. The same instrument and the same technique, applied to a body that is all interior — and the odd harmonics, which a symmetric body cannot have, gave the depth of the winds.

The Moon. Two spacecraft flying in formation measured their separation to microns with a link between them, which is a far more sensitive geometry than a link to the Earth, and produced a lunar gravity field to degree 900.

Two frequencies, and why one is not enough #

The link described above was treated as a single carrier, and a gravity-science spacecraft carries two. The reason is the one term in the error budget that cannot be modelled and can be cancelled.

The solar plasma along the signal path is a dispersive medium: it delays a signal by an amount proportional to the square of the wavelength. Everything else in the chain — the geometry, the troposphere’s wet delay, the spacecraft’s own motion — is non-dispersive and affects both frequencies identically.

So observing at two well-separated frequencies and differencing gives the plasma term directly, and removing it leaves the part that is about the trajectory. That is the same differencing logic that removes the clock offset from an interferometric delay, applied to a different contaminant.

The choice of frequencies follows from the same scaling. A carrier at 32 gigahertz suffers about a fourteenth of the phase noise from plasma that one at 8.4 gigahertz does, so the higher band is where the measurement is made and the lower one is there partly to calibrate it and partly because it is what the rest of the spacecraft’s communications use.

The cost is that a higher frequency needs a tighter beam and therefore better pointing, and it is more strongly attenuated by rain at the ground station. Those are the reasons the higher band was not adopted for everything, and they are why a gravity campaign is scheduled against weather at the receiving complex in a way ordinary tracking is not.

The measurement’s precision is therefore a property of the sky between two points as much as of the equipment at either end, and the term that dominates it varies over the solar cycle.

What the same link measured on the way past the Sun #

There is a result from exactly this apparatus that has nothing to do with any moon, and it is worth recording because it shows what a well-characterised link is worth once it exists.

When a spacecraft passes behind the Sun as seen from the Earth, the radio signal travels through the Sun’s gravitational field, and general relativity says the round-trip time is lengthened by an amount depending on how close the path passes to the Sun. The effect is a delay of about a fifth of a millisecond at grazing incidence, and it is proportional to a parameter that measures how much space-time curvature a unit mass produces — exactly one in general relativity, and adjustable in the alternatives.

Measuring it requires the same things a gravity flyby requires: a coherent two-way link, a model of everything else affecting the path, and a way to remove the plasma. The last of those is the hard part, because the signal is passing through the corona, which is where the plasma is thickest and most variable — so the experiment is done at the one geometry where the dominant error term is at its worst.

The dual-frequency link is what makes it possible, and the result is a confirmation of the prediction to about two parts in a hundred thousand, which stood for two decades as the tightest test of its kind.

That is a measurement of the theory of gravitation, made with equipment built to find out where a spacecraft was, using a calibration developed to weigh a moon. The instrument was a radio link and everything else about it was a choice of what to point it through, which is the most economical description of this whole technique.

The moment the technique became a science instrument #

It is worth recording that gravity science was not designed. Deep-space tracking exists to navigate, and the accuracy it achieves was driven by the need to hit an aim point at a planet after a cruise of years.

The first interior results were by-products. Ranging and Doppler data collected for navigation were reprocessed afterwards, and the masses of the outer planets and their major satellites — quantities that had been known to a per cent from centuries of astrometry — improved by three or four orders of magnitude within a decade of the first flybys. The moment-of-inertia factors that populate the comparison of solar-system interiors mostly came out of that reprocessing.

What changed later was that missions began to be flown for the measurement: trajectories chosen for their gravity yield rather than for their imaging opportunities, encounters scheduled at particular orbital phases, and passes flown with the spacecraft’s own accelerations minimised. The Titan Love number is a result of that second era, and it could not have been extracted from data taken for any other purpose. Both of those results share a property worth naming: the quantity that limited them was the same one, and it was neither the clock nor the antenna. It was the plasma between here and there, which is why the two frequencies are the instrument as much as the dish is.

What the residual cannot say #

It does not see the ocean. What is measured is a deformation larger than a solid can produce. Everything after that — that the deformation is due to a decoupling shell, that the decoupling layer is liquid, that the liquid is water — is inference, of the kind a Love number always requires, resting on the composition being what a body of that density and formation location should be made of.

It does not give a depth. A Love number is one integral of the interior; it constrains a combination of the shell’s thickness, the ocean’s depth and the underlying rigidity, and inverting for any one requires assuming the others. Titan’s ocean depth is quoted with a range of hundreds of kilometres.

It is a snapshot. The measurement is of a response at one frequency, the orbital one. A body whose rheology is frequency-dependent — which is every real body — responds differently at other periods, and nothing in a flyby campaign measures that. The ambiguity between a stiff interior and a soft one is not resolved by a static Love number either. And the tracking geometry itself at a high declination, since what a single pass can separate depends on how much of the sky the antenna sweeps through during it.

Where the ladder goes #

The earlier rungs of this anchor were about navigation: a position from a frequency and an angle from a quasar. Both are about knowing where a spacecraft is. This one is about what the spacecraft is for.

The natural next steps are the ones the missions now in flight will supply. A spacecraft in orbit around an icy moon rather than flying past it can measure the time-variable field over many cycles, separating the tidal response from everything static, and can reach the higher harmonics that say how the shell’s thickness varies. And a link between two spacecraft — the geometry that mapped the Moon — measures a gradient rather than a velocity, which is a far stronger constraint on a small body’s field than anything achievable from the Earth.