An acceleration that was the spacecraft's own heat

Pioneer 10 and Pioneer 11 were tracked by radio for three decades as they left the solar system, and the Doppler residuals of both showed the same thing: an acceleration of about $8.7 \times 10^{-10}$ metres per second squared, directed roughly back towards the Sun, that the navigation model did not account for.

That number is absurdly small. It is a ten-billionth of the acceleration due to gravity at the Earth’s surface. It is also a hundred million times smaller than the solar gravity the spacecraft were already feeling at the distance where it was first measured. Nothing about it should have been detectable at all, and it was detectable because a Doppler measurement integrates: an acceleration that constant, acting for twenty years, moves a spacecraft four hundred thousand kilometres.

Why a residual this small was believable #

A residual is only interesting if the model it is a residual of is good enough. Deep-space navigation is one of the places where that condition is genuinely met.

The observable is a two-way Doppler shift: a signal is transmitted from a station on Earth, coherently transponded by the spacecraft, and compared with the outgoing signal on return. The fractional frequency shift is measured to about $10^{-13}$ over an integration of a thousand seconds, which corresponds to a velocity precision under a tenth of a millimetre a second. Integrated over years, an acceleration of $10^{-10}$ metres per second squared produces a velocity change of a few metres a second, which is four orders of magnitude above the noise.

So the detection was never the difficult part. The difficult part was the list of things that could produce an acceleration that size and were not gravity.

Two further features of the measurement made it hard to dismiss. The first is that both spacecraft showed it, at the same magnitude, while travelling in nearly opposite directions out of the solar system — Pioneer 10 towards the nose of the Sun’s motion through the interstellar medium and Pioneer 11 roughly the other way. A model error tied to a direction would not do that. The second is that the anomaly appeared to switch on: it was not present in the inner solar system, and became measurable somewhere beyond about ten astronomical units. That was for a mundane reason, which is that inside that distance the solar radiation pressure and the ordinary manoeuvring dominate and the residual is buried, but it was widely read as a distance dependence and encouraged the wrong kind of hypothesis.

The spacecraft themselves were also unusually good instruments for this without having been designed to be. Both were spin-stabilised rather than three-axis stabilised, which means they held their attitude without thrusters and therefore without the small unmodelled impulses that attitude control produces. A three-axis spacecraft dumps momentum every few weeks, and each dump is an unmodelled velocity change of exactly the size being looked for here. The Pioneers spun, so the tracking arc was uninterrupted for years at a time.

The list of ordinary explanations, and why it was not obviously closed #

The published analyses worked through the candidates carefully, and the reason the anomaly survived is that the largest candidate was also the hardest to compute.

Solar radiation pressure was the first suspect and it is easy to eliminate: it falls off as the inverse square of the distance, and the residual did not. By twenty astronomical units the sunlight on the dish contributes under one per cent of the anomaly. It has the wrong size and the wrong distance dependence.

Gas leaks from the propulsion system were the most plausible mundane explanation and the least testable. A leak would be episodic and would change when thrusters were fired; the residual was steady. But a slow, steady leak from a sealed line could not be excluded from the outside, and it remained on the list.

Drag against the interplanetary medium is far too small by many orders of magnitude, and it would fall with distance as the density does.

Errors in the ephemeris or in the tracking model were examined at length: an unmodelled mass in the outer solar system, a systematic in the Earth’s own orbit, an error in the time standards. All of these produce signatures with a one-year periodicity, because the observer is on a moving Earth, and the residual had no annual term of the required size.

Thermal recoil was on the list from the beginning, and it was estimated. The estimates disagreed with each other by a factor of several, because the calculation requires knowing where every watt on a complicated spacecraft goes and in which direction it leaves.

That last difficulty deserves more than a clause, because it is the whole reason the episode lasted a decade. Computing a thermal recoil is not a matter of dividing the power by the speed of light. It requires a geometric model of every radiating surface, an emissivity for each, a temperature for each, and — hardest of all — an accounting of the radiation that leaves one surface and strikes another. The Pioneer generators hang on booms about three metres from a 2.7-metre parabolic dish, and a share of their infrared output lands on the back of that dish and is re-emitted or reflected forwards. The size of that share depends on view factors between curved surfaces at awkward angles, and it is the single largest term in the budget. An error of thirty per cent in it is an error of twenty per cent in the answer, and nobody had a reason to have computed those view factors to better than that before the anomaly existed.

The test that should have settled it #

There was one clean discriminant available, and it was applied, and it was not decisive.

Heat from a radioisotope source declines. Plutonium-238 has a half-life of 87.7 years, so the thermal output of the generators fell by about eight per cent over the eleven-year span of the best-analysed data. An acceleration produced by that heat must fall by the same eight per cent. An acceleration produced by a modification of gravity, or by anything else that does not care about the plutonium, must not.

That is a frustrating place for a measurement to sit and it is a common one. The test is correct in principle, the effect is present, and the signal-to-noise is a little under one. Repeating the analysis with different data cuts moved the answer around inside its error bar, which is exactly what a marginal result does.

There is a general point hiding in that list, and it is the reason the elimination took so long. Every candidate on it was rejected by a signature rather than by a size — the wrong distance dependence, the wrong time dependence, the wrong periodicity. Thermal recoil was the one candidate whose signature was almost exactly right: constant in direction, nearly constant in time, and with a magnitude that could plausibly be anything within a factor of three of the observed value. A hypothesis that cannot be distinguished from the observation by its shape can only be tested by computing its size, and computing its size was the part nobody could do.

What was actually measured, in the end #

The resolution came from an unglamorous source: the spacecraft’s own telemetry.

Both Pioneers had transmitted housekeeping data continuously — temperatures at dozens of points on the structure, electrical currents, generator output — and those records had been archived on magnetic tape and largely forgotten. Recovering them, and reconstructing the file formats, took years. What they provided was the one thing every previous thermal estimate had lacked: a measured temperature at each surface at each epoch, rather than a modelled one.

With the temperatures known, the calculation becomes a finite-element radiative transfer problem on a known geometry with no free parameters. Two independent groups did it, using different codes and different treatments of the reflection off the dish, and both found a thermal recoil that accounts for the whole anomaly within the uncertainties. The residual after subtracting the modelled thermal force is consistent with zero.

The decay test then becomes a consistency check rather than the primary evidence, and it passes: the reconstructed thermal acceleration declines at the rate the telemetry says the power declined, and the tracking residuals, though noisy, prefer the declining model when it is fitted with the amplitude fixed by the thermal calculation rather than left free.

It is worth recording how much of the recovered signal was in the tapes rather than in the model. The reconstruction used more than thirty years of telemetry from both spacecraft, comprising millions of individual measurements, and the crucial quantities were the temperatures of the generator fins and of the compartment walls. Those temperatures had been transmitted for engineering reasons and nobody had ever needed them. The lesson that the mission operators drew was not about gravity: it was that housekeeping telemetry is science data, and that archiving it in a readable format costs nothing compared with what it can later settle.

Where the picture stops #

The picture stops in three places, and the first is a caution about the moral rather than about the physics.

It is not a story about scientists chasing new physics carelessly. The papers that reported the anomaly stated repeatedly that thermal recoil was the leading candidate and that it could not be computed to better than a factor of two with the information available. The proposals to modify gravity came from elsewhere, and they were mostly written by people who had read the abstract. The measurement itself was careful, was published with its systematic budget, and turned out to be right about everything except the interpretation it explicitly declined to make.

A thermal force cannot be eliminated by design, only reduced. Any spacecraft that dissipates power radiates it, and any spacecraft that is not spherically symmetric radiates it anisotropically. What can be done is to make the geometry simple enough to compute: the later missions built for precise tracking use spherical or highly symmetric enclosures and instrument the temperatures densely, so that the recoil is a modelled term with a small uncertainty rather than an unmodelled one with a large one.

And the residual was not zero before the thermal model, which matters. A model with a missing term does not merely have a larger error bar; it has a biased answer, and the bias is absorbed into whatever parameter can accommodate it. In this case the accommodation was a constant acceleration, which is a parameter the fit happened to have. Had the tracking geometry been slightly different, the same missing heat would have been absorbed into a mass, a position or a range bias, and nothing anomalous would have been reported at all.

There is also a fourth limit worth stating separately because it is about what the episode does and does not license. The anomaly is explained, and that is a statement about this measurement rather than about the general question. Tests of gravity at low accelerations continue, and they continue for a good reason: the acceleration scale where the anomaly sat is coincidentally close to the scale at which galactic rotation curves start to disagree with Newtonian expectations, and that coincidence is why the Pioneer result attracted the attention it did. Resolving the Pioneer number does not resolve that other question. It removes one piece of evidence that was never very good, and it does so in a way that leaves the interesting question exactly where it was.

The same physics, four orders of magnitude larger #

The mechanism here is not exotic and the collection has already met it. An asteroid’s orbit is moved by heat through exactly this recoil: sunlight is absorbed on the morning side, re-radiated with a lag from the afternoon side, and the imbalance pushes. A comet arriving a day early is the same effect with sublimating ice doing the pushing rather than infrared photons, and the drift rate of a small asteroid says what its surface is made of because the lag depends on the thermal inertia.

The parallel goes further than the mechanism. In both cases the acceleration is proportional to the radiated power and inversely proportional to the mass, so it matters for small things and not for large ones: the same recoil on a kilometre-sized asteroid moves its semi-major axis by metres a year, and on a hundred-metre one by hundreds of metres a year. A spacecraft is at the extreme end of that scaling — a couple of hundred kilograms with kilowatts of internal power — which is why an effect that is a curiosity for a planet is a first-order term for a probe.

What is different about the spacecraft case is the sign of the ignorance. For an asteroid, the thermal properties are unknown and the drift is measured, so the drift is used to infer the surface. For a spacecraft, the thermal properties are documented and the drift is measured, so the two can be compared — and when they disagree, the disagreement is a statement about the bookkeeping.

Why a point mass is never the right model #

The general lesson is one the navigation essays keep arriving at from different directions. A trajectory model treats the spacecraft as a point with a mass and a cross-section. Every real spacecraft is an object with a shape, a temperature distribution, an attitude, and articulated parts, and each of those contributes an acceleration at the $10^{-10}$ level or above.

For most missions that does not matter, because the required accuracy is orders of magnitude coarser. It matters exactly when the tracking is being used to measure something rather than to arrive somewhere: an ocean inferred from a Doppler residual, a gravity field mapped from a low orbit, a test of a relativistic parameter. In each of those the science is in the residual, and every unmodelled force on the spacecraft is a competing explanation for it.

The Pioneer episode is the cleanest available demonstration of that, precisely because the answer turned out to be mundane. A thirty-year, twenty-astronomical-unit measurement of unprecedented precision detected an acceleration that was really there, and it was the spacecraft’s own waste heat, documented in a filing cabinet, radiating one part in forty more in one direction than the other.

One more consequence is worth drawing out, because it is the practical residue of the whole affair. The way a small unmodelled force enters a trajectory fit is through the equations that say which component of an acceleration moves which orbital element, and those equations are why a constant radial force on an escaping spacecraft is so easy to detect and so hard to attribute. A force along the line of sight from the Sun does no work on a body moving nearly radially away from it, but it changes the range at every epoch, which is precisely the quantity a Doppler measurement integrates. The same force applied to a body on a circular orbit would be almost invisible in range and would show up instead as a slow change in the period. Detectability is a property of the geometry as much as of the force, and the Pioneers were in the one configuration that made this particular force maximally visible.

A last observation about the episode’s shape. The anomaly was not resolved by a new measurement of the spacecraft — it was resolved by recovering telemetry nobody had kept for the purpose and building a thermal model of a vehicle that had been out of contact for years. The evidence had existed all along in a form nobody had thought to treat as evidence, which is the ordinary way an old systematic is finally closed.

Where the ladder goes next #

The next rung is the family of non-gravitational forces that are designed rather than accidental — the station-keeping budget that a satellite pays every year to fight a force nobody can switch off. The rung after that is the reverse inference: what a spacecraft’s unmodelled acceleration can be made to measure once it is understood, which is how a probe becomes an instrument for the field it is falling through.