1 Introduction

For generations of researchers our solar system provided opportunities to establish and test fundamental laws of gravity. By studying the motion of planets, their moons, and comets, astronomers learned the basic rules that govern the dynamics of a system of gravitating bodies. Today we apply this knowledge to study the universe around us, expecting that the same laws of gravity govern the behavior of the universe on large scales, from planetary systems similar to our own to galaxies and to the entire cosmos as a whole.

Astronomers, however, do not normally discover new laws of nature. We are not yet able to manipulate the objects of our scrutiny. The telescopes and detectors we operate are simply passive probes that cannot order the cosmos what to do. Yet they can tell us when something isn’t following established rules. For example, take the planet Uranus, whose discovery is credited to the English astronomer William Herschel and dated to 1781 (others had already noted its presence in the sky but misidentified it as a star). As observational data about its orbit accumulated over the following decades, people began to notice that Uranus’s orbit deviated slightly from the dictates of Newton’s gravity, which by then had withstood a century’s worth of testing on the other planets and their moons. Some prominent astronomers suggested that perhaps Newton’s laws begin to break down at such great distances from the Sun.

This led immediately to the question: What is there to do? Abandon or modify Newton’s laws and come up with new rules of gravity? Or postulate a yet-to-be-discovered planet in the outer solar system, whose gravity was absent from the calculations for Uranus’s orbit? The answer came in 1846, when astronomers discovered the planet Neptune just where a planet had to be for its gravity to perturb Uranus in just the ways measured. Newton’s laws were safe…for the time being.

Then there is Mercury, the planet closest to the Sun. Its orbit, too, habitually disobeyed Newton’s laws of gravity resulting in an anomalous precession of its perihelion. This anomaly was known for a long time; it amounts to 43 seconds of arc (”) per century and cannot be explained within Newton’s gravity, thereby presenting a challenge for physicists and astronomers. In 1855, the French astronomer Urbain Jean Joseph Le Verrier, who in 1846 predicted Neptune’s position in the sky within one degree, wrote that the anomalous residue of the Mercurial precession would be accounted for if yet another as-yet undiscovered planet — call it Vulcan — revolves inside the Mercurial orbit so close to the Sun that it would be practically impossible to discover in the solar glare, or perhaps it was an entire uncatalogued belt of asteroids orbiting between Mercury and the Sun.

It turns out that Le Verrier was wrong on both counts. This time he really did need a new understanding of gravity. Within the limits of precision that our measuring tools impose, Newton’s laws behave well in the outer solar system. However, they break down in the inner solar system, where the Sun’s gravitational field is so powerful that it warps space. And that is where we cannot ignore the effects of general relativity. It took another 60 years to solve this puzzle. In 1915, before publishing the historical paper with the field equations of the general theory of relativity (e.g., [115, 116]), Albert Einstein computed the expected perihelion precession of Mercury’s orbit. This was not the first time Einstein tackled this problem: indeed, earlier versions of his gravity theory were rejected, in part, because they predicted the wrong value (often with the wrong sign) for Mercury’s perihelion advance [112]. However, when he obtained the famous 43”/century needed to account for the anomaly, he realized that a new era in gravitational physics had just begun.

The stories of these two planets, Mercury and Uranus, involve two similar-looking anomalies, yet two completely different solutions.

Ever since its original publication on November 25, 1915 [115, 116], Einstein’s general theory of relativity continues to be an active area of both theoretical and experimental research [388, 389]. Even after nearly a century since its discovery, the theory successfully accounts for all solar system observations gathered to date; it is remarkable that Einstein’s theory has survived every test [418]. In fact, both in the weak field limit evident in our solar system and with the stronger fields present in systems of binary pulsars the predictions of general relativity have been extremely well tested. Such longevity and success make general relativity the de facto “standard” theory of gravitation for all practical purposes involving spacecraft navigation, astronomy, astrophysics, cosmology and fundamental physics [399].

Remarkably, even after more than 300 years since the publication of Newton’s “Principia” and nearly 100 years after the discovery of Einstein’s general theory of relativity, our knowledge of gravitation is still incomplete. Many challenges remain, leading us to explore the physics beyond Einstein’s theory [388, 389]. In fact, growing observational evidence points to the need for new physics. Multiple dedicated efforts to discover new fundamental symmetries, investigations of the limits of established symmetries, tests of the general theory of relativity, searches for gravitational waves, and attempts to understand the nature of dark matter were among the topics that had been the focus of scientific research at the end of the last century. These efforts have further intensified with the unexpected discovery in the late 1990s of a small acceleration rate of our expanding Universe, which triggered many new activities aimed at answering important questions related to the most fundamental laws of Nature [389, 399].

Many modern theories of gravity that were proposed to address the challenges above, including string theory, supersymmetry, and brane-world theories, suggest that new physical interactions will appear at different ranges. For instance, this may happen because at sub-millimeter distances new dimensions can exist, thereby changing the gravitational inverse-square law [36, 37]. Similar forces that act at short distances are predicted in supersymmetric theories with weak scale compactifications [33], in some theories with very low energy supersymmetry breaking [105], and also in theories of very low quantum gravity scale [109, 304, 366].

Although much of the research effort was devoted to the study of the behavior of gravity at very short distances, notably on millimeter-to-micrometer ranges, it is possible that tiny deviations from the inverse-square law occur at much larger distances. In fact, there is a possibility that noncompact extra dimensions could produce such deviations at astronomical distances [110]. By far the most stringent constraints to date on deviations from the inverse-square law come from very precise measurements of the Moon’s orbit about the Earth. Analysis of lunar laser ranging data tests the gravitational inverse-square law on scales of the Earth-Moon distance [421], so far reporting no anomaly at the level of accuracy of 3 × 10−11 of the gravitational field strength.

While most of the modern experiments in the solar system do not show disagreements with general relativity, there are puzzles that require further investigation. One such puzzle was presented by the Pioneer 10 and 11 spacecraft. The radiometric tracking data received from these spacecraft while they were at heliocentric distances of 20–70 astronomical units (AU) have consistently indicated the presence of a small, anomalous, Doppler frequency drift. The drift was interpreted as a constant sunward acceleration of aP = (8.74 ± 1.33) × 10−10 m/s2 experienced by both spacecraft [24, 27, 390]. This apparent violation of the inverse-square law has become known as the Pioneer anomaly; the nature of this anomaly remains unexplained.

Before Pioneer 10 and 11, Newtonian gravity was not measured with great precision over great distances and was therefore never confirmed. The unique “built-in” navigation capabilities of the two Pioneers allowed them to reach the levels of ∼ 10−10 m/s2 in acceleration sensitivity. Such an exceptional sensitivity allowed researchers to use Pioneer 10 and 11 to test the gravitational inverse square law in the largest-scale gravity experiment ever conducted. However, the experiment failed to confirm the validity of this fundamental law of Newtonian gravity in the outer regions of the solar system. Thus, the nagging question remains: Just how well do we know gravity?

One can demonstrate that beyond 15 AU the difference between the predictions of Newton and Einstein are negligible. So, at the moment, two forces seem to be at play in deep space: Newton’s law of gravity and the Pioneer anomaly. Until the anomaly is thoroughly accounted for by conventional causes, and can therefore be eliminated from consideration, the validity of Newton’s laws in the outer solar system will remain in doubt. This fact justifies the importance of the investigation of the nature of the Pioneer anomaly.

However, the Pioneer anomaly is not the only unresolved puzzle. Take the dark matter and dark energy problem. While extensive efforts to detect the dark matter that is believed to be responsible for the puzzling observations of galaxy rotation curves have not met with success so far, modifications of gravitational laws have also been proposed as a solution to this puzzle. We still do not know for sure whether or not the ultimate solution for the dark matter problem will require a modification of the Standard Model of cosmology, but some suggested that new gravitational laws are at play in the arms of spiral galaxies.

A similar solution was proposed to explain the cosmological observations that indicate that the expansion of the universe is accelerating. There is now a great deal of evidence indicating that over 70% of the critical density of the universe is in the form of a “negative-pressure” dark energy component; we have no understanding of its origin or nature. Given the profound challenge presented by the dark energy problem, a number of authors have considered the possibility that cosmic acceleration is not due to a particular substance, but rather that it arises from new gravitational physics (see discussion in [389]).

Many of the models that were proposed to explain the observed acceleration of the universe without dark energy or the observed deviation from Newtonian laws of gravity in the arms of spiral galaxies without dark matter may also produce measurable gravitational effects on the scale of the solar system. These effects could manifest themselves as a “dark force”, similar to the one detected by the Pioneer 10 and 11 spacecraft. Some believe that the Pioneer anomaly may be a critical piece of evidence as it may indicate a deviation from Einstein’s gravity theory on the scales of the solar system. But is it? Or can the Pioneer anomaly be explained by the mundane physics of a previously unaccounted-for on-board systematic effect? In this review we summarize the current knowledge of the anomaly and explore possible ways to answer this question.

The review is organized as follows. We begin with descriptions of the Pioneer 10 and 11 spacecraft and the strategies for obtaining and analyzing their data. In Section 2 we describe the Pioneer spacecraft. We provide a significant amount of information on the design, operations and behavior of Pioneer 10 and 11 during their entire missions, including information from original project doc umentation, descriptions of various data formats, and techniques used for their navigation. This information is critical to the ongoing investigation of the Pioneer anomaly.

In Section 3 we describe the techniques used for acquisition of the Pioneer data. In particular, we discuss the Deep Space Network (DSN), its history and current status, describe the DSN tracking stations and details of their operations in support of deep space missions. We present the available radiometric Doppler data and describe techniques for data preparation and analysis. We also discuss the Pioneer telemetry data and its value for the anomaly investigation.

In Section 4 we address the basic elements of the theoretical foundation for precision spacecraft navigation. In particular, we discuss the observational techniques and physical models that were used for precision tracking of the Pioneer spacecraft and analysis of their data. We describe models of gravitational forces and those that are of nongravitational nature.

In Section 5 we focus on the detection and initial characterization of the Pioneer anomaly. We describe how the anomalous acceleration was originally identified in the data. We continue by summarizing the current knowledge of the physical properties of the Pioneer anomaly. We briefly review the original efforts to understand the signal.

In Section 6 we review various mechanisms proposed to explain the anomaly that use unmodeled forces with origin either external to the spacecraft or those generated on-board. We discuss theoretical proposals that include modifications of general relativity, modified Newtonian gravity, cosmological theories, theories of dark matter and other similar activities. We review the efforts at independent confirmation with other spacecraft, planets, and other bodies in the solar system.

In Section 7, we describe the results of various independent studies of the Pioneer anomaly. We discuss new Pioneer 10 and 11 radiometric Doppler data that recently became available. This much extended set of Pioneer Doppler data is the primary source for the new investigation of the anomaly. A near complete record of the flight telemetry that was received from the two Pioneers is also available. Together with original project documentation and newly developed software tools, this additional information is now used to reconstruct the engineering history of both spacecraft, with special emphasis on the possible contribution to the anomalous acceleration by on-board systematic effects. We review the current status of these efforts to investigate the anomaly.

In Section 8, we present our summary and conclusions.

In Appendices AD, we provide additional information on the geometry and design of Pioneer 10 and 11 spacecraft and describe various data formats used for mission operations.

2 The Pioneer 10 and 11 Project

NASA’s Pioneer program began in 1958, in the earliest days of the space age, with experimental spacecraft that were designed to reach Earth escape velocity and perform explorations of the interplanetary space beyond the Earth’s orbit. Several of these launch attempts ended in failure; the five spacecraft that reached space later became known as Pioneers 1–5. These were followed in the second half of the 1960s by Pioneers 6–9, a series of significantly more sophisticated spacecraft that were designed to be launched into solar orbit and make solar observations. These spacecraft proved extremely robustFootnote 1 and paved the way for the most ambitious projects yet in the unmanned space program: Pioneers 10 and 11.

The Pioneer 10 and 11 spacecraft were the first two man-made objects designed to explore the outer solar system (see details in [294, 291, 293, 290, 292, 289, 162, 295, 286, 354, 285, 127, 126, 143, 272, 350, 296, 383, 385, 386]). Their objectives were to conduct, during the 1972–73 Jovian opportunities, exploratory investigation beyond the orbit of Mars of the interplanetary medium, the nature of the asteroid belt, the environmental and atmospheric characteristics of Jupiter and Saturn (for Pioneer 11), and investigate the solar system beyond Jupiter’s orbit.

In this section we review the Pioneer 10 and 11 missions. We present information about the spacecraft design. Our discussion focuses on subsystems that played important roles in the continued functioning of the vehicles and on subsystems that may have affected their dynamical behavior: specifically, we review the propulsion, attitude control, power, communication, and thermal subsystems. We also provide information about the history of these systems throughout the two spacecrafts’ exceptionally lengthy missions.

2.1 The Pioneer 10 and 11 missions

The Pioneer missions were the first to cross the asteroid belt, perform in situ observations of the interplanetary medium in the outer solar system, and close-up observations of the gas giant Jupiter. Their mission design was characterized by simplicity: a powerful launch vehicle placed the spacecraft on an hyperbolic trajectory aimed directly at Jupiter, which the spacecraft were expected to reach approximately 21 months after launch.

At the distance of Jupiter, operating a spacecraft using solar panels is no longer practical (certainly not at the level of technology that was available to the designers of the Pioneer missions in the late 1960s.) For this reason, nuclear power was chosen as the means to provide electrical power to the spacecraft, in the form of 238Pu powered radioisotope thermoelectric generators (RTGs). As even this was relatively new technology at the time the missions were designed, the power subsystem was suitably over-engineered, the design requirement being a completely functional spacecraft capable of performing all planned science observations with only three (out of four) RTGs operating.

Such conservative engineering characterized the entire design of these spacecraft and their missions, and it was likely responsible for the two spacecrafts’ exceptional longevity, and their ability to deliver science results that far exceeded the expectations of their designers. The original plan envisioned a primary mission of 600–900 days in duration. Nevertheless, following its encounter with Jupiter, Pioneer 10 remained functional for over 30 years; meanwhile Pioneer 11, though not as long lived as its sister craft, successfully navigated a path across the solar system for another encounter with Saturn, offering the first close-up observations of the ringed planet.

After the Jupiter and Saturn (for Pioneer 11) encounters (see Figure 2.1), the craft followed escape hyperbolic orbits near the plane of the ecliptic on opposite sides of the solar system, continuing their extended missions [126]. (See Figure 2.2.) The spacecraft explored the outer regions of the solar system, studying energetic particles from the Sun (solar wind), and cosmic rays entering our portion of the Milky Way. Major milestones of the two Pioneer projects are shown in Table 2.1.

Figure 2.1
figure 1

Trajectories of Pioneer 10 and 11 during their primary missions in the solar system (from [126]). The time ticks shown along the trajectories and planetary orbits represent the distance traveled during each year.

Figure 2.2
figure 2

Ecliptic pole view of the Pioneer 10 and Pioneer 11 trajectories during major parts of their extended missions. Pioneer 10 is traveling in a direction almost opposite to the galactic center, while Pioneer 11 is heading approximately in the shortest direction to the heliopause. The direction of the solar system’s motion in the galaxy is approximately towards the top. (From [27].)

Table 2.1 Major milestones of the Pioneer 10 and 11 projects.

The Pioneers were excellent vehicles for the purposes of precision celestial mechanics experiments [24, 27, 32, 194, 255, 256, 257, 259, 260, 262, 263, 274, 390, 391, 392, 393]. This was due to a combination of many factors, including the presence of a coherent mode transceiver on board, their attitude control (spin-stabilized, with a minimum number of attitude correction maneuvers using thrusters), power design (the RTGs being on extended booms aided the stability of craft and also reduced thermal effects on the craft; see Figure 2.3), and precise Doppler tracking (with the accuracy of post-fit Doppler residuals at the level of mHz). The exceptional “built-in” acceleration sensitivity of the Pioneer 10 and 11 spacecraft naturally allowed them to reach a level of accuracy of ∼ 10−10 m/s2. The result was one of the most precise spacecraft navigations in deep space to date [269].

Figure 2.3
figure 3

A drawing of the Pioneer spacecraft. (From [292].)

2.1.1 Pioneer 10 mission details

Pioneer 10 was launched on 2 March 1972 (3 March 1972 at 01:49 Universal Coordinated Time) from Cape Canaveral on top of an Atlas/Centaur/TE364-4 launch vehicle [190]. The launch marked the first use of the Atlas-Centaur as a three-stage launch vehicle.

The launch vehicle configuration was an Atlas launch vehicle equipped with a Centaur D upper stage and a TE364-4 solid-fuel third stage that provided additional thrust and also supplied the initial spin of the spacecraft. The third stage was required to accelerate Pioneer 10 to the speed of 14.39 km/s, needed for the flight to Jupiter.

After a powered flight of approximately 14 minutes, the spacecraft was separated from its launch vehicle; its initial spin ∼ 60 revolutions per minute (rpm) was reduced by thrusters, and then reduced further when the magnetometer and RTG booms were extended [283, 145]. The spacecraft was then oriented to ensure that its high-gain antenna pointed towards the Earth. Thus, the initial cruise phase from the Earth to Jupiter began.

The first interplanetary cruise phase of Pioneer 10 took approximately 21 months. During this time, Pioneer 10 successfully crossed the asteroid belt, demonstrating for the first time that this region of the solar system is safe for spacecraft to travel through.

Pioneer 10 arrived at Jupiter in late November, 1973 [162]. Its closest approach to the red giant occurred on 4 December 1973, at 02:25 UTC. It performed the first ever close-up observations of the gas giant, before continuing its journey out of the solar system on an hyperbolic escape trajectory (Figure 2.2). During the planetary encounter, Pioneer 10 took several photographs of the planet and its moons, measured Jupiter’s magnetic fields, and observed the planet’s radiation belts. Radiation in the Jovian environment, potentially damaging to the spacecraft’s electronics, was a concern to the mission designers. However, Pioneer 10 survived the planetary encounter without significant damage, although its star sensor became inoperative shortly afterwards [349], a likely result of excessive radiation exposure near Jupiter.

The encounter with Jupiter changed Pioneer 10’s trajectory as was planned by JPL navigators [404]. As a result, Pioneer 10 was now on an hyperbolic escape trajectory that took it to ever more distant parts of the solar system. Originally, signal loss was expected before Pioneer 10 reached twice the heliocentric distance of Jupiter (the downlink telecommunication power margin was 6 dB at the time of Jupiter encounter); however, continuing upgrades to the facilities of the Deep Space Network (DSN) permitted tracking of Pioneer 10 until the official termination of Pioneer 10’s science mission in 1997 and even beyond.

Pioneer 10 continued to make valuable scientific investigations until its science mission ended on March 31, 1997. After this date, Pioneer 10’s weak signal was tracked by the NASA’s DSN as part of an advanced concept study of communication technology in support of NASA’s future interstellar probe mission. Pioneer 10 eventually became the first man-made object to leave the solar system.

During one of the last attempts to contact Pioneer 10, in April 2001, at first no signal was detected; however, the spacecraft’s signal did appear once it detected a signal from the Earth and its radio system switched to coherent mode. From this, it was concluded that the on-board transmitter frequency reference (temperature controlled crystal oscillator) failed, possibly due to the combined effects of aging, the extreme cold environment of deep space, and a drop in the main bus voltage due to the depletion of the spacecraft’s RTG power source. This failure had no impact on the ability to obtain precision Doppler measurements from Pioneer 10.

On March 2, 2002 NASA’s DSN made another contact with Pioneer 10 and confirmed that the spacecraft was still operational thirty years after its launch on March 3, 1972 (UT). The uplink signal was transmitted on March 1 from the DSN’s Goldstone, California facility and a downlink response was received twenty-two hours later by the 70-meter antenna at Madrid, Spain. At this time the spacecraft was 11.9 billion kilometers from Earth at about 79.9 AU from the Sun and heading outward into interstellar space in the general direction of Aldebaran at a distance of about 68 light years from the Earth, and a travel time of two million years. The last telemetry data point was obtained from Pioneer 10 on 27 April 2002 when the craft was 80 AU from the Sun.

The last signal from Pioneer 10 was received on Earth on 23 January 2003, when NASA’s DSN received a very weak signal from the venerable spacecraft from the distance of ∼ 82.1 AU from the Sun. The previous three contacts had very faint signals with no telemetry received. At that time, NASA engineers reported that Pioneer 10’s RTG has decayed to the point where it may not have enough power to send additional transmissions to Earth. Consequently, the DSN did not detect a signal during a contact attempt on 7 February 2003. Thus, after more than 30 years in space, the Pioneer 10 spacecraft sent its last signal to Earth.

The final attempt to contact Pioneer 10 took place on the 34th anniversary of its launch, on 3–5 March 2006 [397]. At that time, the spacecraft was 90.08 AU from the Sun, moving at 12.08 km/s. The round-trip light time (i.e., time needed for a DSN radio signal to reach Pioneer 10 and return back to the Earth) was approximately 24 h 56 m, so the same antenna, DSS-14 at Goldstone, CA, was used for the track. Unfortunately, no signal was received. Given the age of the spacecraft’s power source, it was clear that there was no longer sufficient electrical power on board to operate the transmitter [378].

2.1.2 Pioneer 11 mission details

Pioneer 11 followed its older sister approximately one year later. It was launched on 5 April 1973 (on April 6, 1973 at 02:11 UTC), also on top of an Atlas/Centaur/TE364-4 launch vehicle. The second stage used for Pioneer 11 was a Centaur D-1A, while the third stage was a TE364-4 solid fuel vehicle.

After safe passage through the asteroid belt on 19 April 1974, Pioneer 11’s thrusters were fired to add another ∼ 65 m/s to the spacecraft’s velocity. This adjusted the aiming point at Jupiter to 43,000 km above the cloud tops. The close approach also allowed the spacecraft to be accelerated by Jupiter to a velocity of 48.06 km/s, so that it would be carried across the solar system some 2.4 billion km to Saturn.

Early in its mission, Pioneer 11 suffered a propulsion system anomaly that caused the spin rate of the spacecraft to increase significantly (see Figure 2.16). Fortunately, the spin rate was not high enough to endanger the spacecraft or compromise its mission objectives.

Pioneer 11’s first interplanetary cruise phase lasted approximately 20 months. During this time, a major trajectory correction maneuver was performed, aiming Pioneer 11 for a precision encounter with Jupiter. Pioneer 11’s closest approach to Jupiter occurred on 2 December 1974 at 17:22 UTC. This encounter provided the necessary gravity assist to alter Pioneer 11’s trajectory for a planned encounter with Saturn (see Figure 2.1).

The second interplanetary cruise phase of Pioneer 11’s mission took it across the solar system. Initially, Pioneer 11’s heliocentric distance was actually decreasing as it followed an hyperbolic trajectory taking the spacecraft more than 1 AU above the plane of the ecliptic. This phase of the mission culminated in a successful encounter with Saturn. Pioneer 11’s closest approach to the ringed planet occurred on 1 September 1979, at 16:31 UTC. Still fully operational, Pioneer 11 was able to make close-up observations of the ringed planet.

After this second planetary encounter, Pioneer 11 continued to escape the solar system on an hyperbolic escape trajectory, and remained operational for many years. Pioneer 11 explored the outer regions of our solar system, studying the solar wind and cosmic rays.

The spacecraft sent its last coherent Doppler data on October 1, 1990 while at 31.7 AU from the SunFootnote 2. In October 1990 a microwave relay switch failed on board Pioneer 11, in its communications subsystem. The most notable consequence of this failure is that it was no longer possible to operate this spacecraft’s radio system in coherent mode, which is required for precision Doppler observations. Therefore, after this event, precision Doppler data was no longer produced by the Pioneer 11 spacecraft.

The spacecraft continued to provide science observations until the end of its mission in 1995. In September 1995, Pioneer 11 was at a distance of 6.5 billion km from Earth. At that distance, it takes over 6 hours for the radio signal to reach Earth. However, by September 1995, Pioneer 11 could no longer make any scientific observations as its power supply was nearly depleted. On 30 September 1995, routine daily mission operations were stopped. Intermittent contact continued until November 1995, at which time the last communication with Pioneer 11 took place. There has been no communication with Pioneer 11 since. The Earth’s motion has carried our planet out of the view of the spacecraft antenna.

2.1.3 Pioneer 10 and 11 project documentation

Up until 2005, very little documentation on the Pioneer spacecraft was available to researchers. Indeed, around this time much of the Pioneer archival material stored at NASA’s Ames Research Center was scheduled for destruction due to budget constraints.

The growing interest in the Pioneer anomaly helped to initiate an effort at the NASA Ames Research Center to recover the entire archive of the Pioneer Project documents for the period from 1966 to 2003 (see details in [379, 397]). This massive archive contains all Pioneer 10 and 11 project documents discussing the spacecraft and mission design, fabrication of various components, results of various tests performed during fabrication, assembly, pre-launch, as well as calibrations performed on the vehicles; and also administrative documents including quarterly reports, memoranda, etc. Most of the maneuver records, spin rate data, significant events of the craft, etc., have also been identified.

A complete set of Pioneer-related documentation is listed in the Bibliography. Here, we mention some of the more significant pieces of documentation that are essential to understanding the Pioneer 10 and 11 spacecraft and their anomalous accelerations:

  • The first document to be mentioned is entitled “Pioneer F/G: Spacecraft Operational Characteristics” [292] (colloquially referred to by its identifier as “PC-202”), and contains a complete description of the Pioneer 10 and 11 spacecraft and their subsystems. The document was last revised in mid-1971, just months before the launch of Pioneer 10, indicating that it reflects accurately the configuration of the Pioneer 10 spacecraft as it flew.

  • Valuable information was found in the TRW Systems Group’s document entitled “Pioneer Project Flights F and G Final Report” [386], which contained post-launch information about both spacecraft, including, among other things, detailed information about their exact launch configuration.

  • Details about the SNAP-19 radioisotope thermoelectric generators can be found in Teledyne Isotopes Energy Systems Division’s “SNAP 19 Pioneer F & G Final Report” [350], which has been released for public distribution on 9 February 2006.

  • Much additional detail about the thermal design of the spacecraft was obtained from TRW Systems Group’s “Pioneer F/G Thermal Control Subsystem Design Review Number 3” [385].

  • The Master Data Record (MDR) file format is described in Alliedsignal Technical Services Corporation’s document [402]. Sensor calibration data for Pioneer 10 is provided by BENDIX Field Engineering Corporation [370]; for Pioneer 11, the same information was provided privately by L.R. Kellogg [167]. The data format used by scientific instruments in the MDRs is described in “Pioneer F/G: EGSE Computer Programming Specifications for Scientific Instruments” [289]; further information about both scientific and engineering data words is present in “Pioneer F/G: On-line Ground Data System Software Specification” [291]. Together, these resources make it possible to read and interpret the entire preserved telemetry record of Pioneer 10 and 11 using modern software [397].

  • Operational details about the Pioneer 10 and 11 missions were recorded in meeting presentations, many of which have been preserved [11, 12, 13, 14, 15, 16, 17, 18]. Additional details are provided in [287, 288, 297, 298, 111, 403].

  • Details about tracking and data acquisition were published in the JPL Deep Space Network’s Technical Reports series [5, 6, 7, 8, 9, 45, 46, 47, 48, 49, 66, 74, 75, 76, 147, 149, 150, 161, 182, 183, 215, 216, 217, 218, 219, 220, 221, 222, 223, 224, 225, 226, 227, 228, 248, 249, 250, 251, 275, 281, 307, 313, 314, 334, 335, 336, 337, 338, 339, 340, 341, 342, 343, 344, 345, 347, 362, 406, 408]. Further, relevant details can be found in [84, 102, 282, 184, 185, 236, 369, 409]. Details about orbit estimation procedures are provided in [58, 119, 131, 132, 133, 134, 136, 372, 425].

The on-going study of the Pioneer anomaly would not be possible without these resources, much of which was preserved only because of the labors of dedicated individuals.

2.2 The Pioneer spacecraft

In this section we discuss the details of the Pioneer 10 and 11 spacecraft, focusing only on those most relevant to the study of the Pioneer anomaly.

2.2.1 General characteristics

Externally, the shape of the Pioneer 10 and 11 spacecraft was dominated by the large (2.74 m diameter) high-gain antenna (HGA)Footnote 3, behind which most of the spacecrafts’ instrumentation was housed in two adjoining hexagonal compartments (Figure 2.3). The main compartment, in the shape of a regular hexagonal block, contained the fuel tank, electrical power supplies, and most control and navigation electronics. The adjoining compartment, shaped as an irregular hexagonal block, contained science instruments. Several openings were provided for science instrument sensors. The internal arrangement of spacecraft components is shown in Figure 2.4.

Figure 2.4
figure 4

Pioneer 10 and 11 internal equipment arrangement. (From [292].)

The main and science compartments collectively formed the spacecraft body, which was covered by multilayer thermal insulation on all sides except part of the aft side, where a passive thermal control louver system was situated.

For the purposes of attitude control, the entire spacecraft was designed to spin in the plane of the HGA.

Three extensible booms were attached to the main compartment, spaced at 120°. Two of these booms, both approximately 3 m long, each held two radioisotope thermoelectric generators (RTGs). This design was dictated mainly by concerns about the effects of radiation from the RTGs on the spacecrafts’ instruments, but it also had the beneficial side effect of minimizing radiative heat exchange between the RTGs and the spacecraft body. The third boom, approximately 6 m in length, held the magnetometer sensor. The length of this boom ensured that the sensor was not responding to magnetic fields originating in the spacecraft itself.

The total mass of Pioneer 10 and 11 was approximately 260 kg at the time of launch, of which approximately 30 kg was propellant and pressurant. The masses of the spacecraft slowly varied throughout their missions primarily due to propellant usage (for details, see Section 2.3.2).

The propulsion system was designed to perform three types of maneuvers: spin/despin (setting the initial spin rate shortly after launch), precession (to keep the HGA pointing towards the Earth, and also to orient the spacecraft during orbit correction maneuvers) and velocity changes. Of the two spacecraft, Pioneer 11 used more of its propellant, in the course of velocity correction maneuvers that were used to adjust the spacecraft’s trajectory for its eventual encounter with Saturn.

2.2.2 Science instruments

The Pioneer spacecraft carried an identical set of 11 science instruments, with a 12th instrument present only on Pioneer 11, namely:

  1. 1.

    JPL Helium Vector Magnetometer

  2. 2.

    ARC Plasma Analyzer

  3. 3.

    U/Chicago Charged Particle Experiment

  4. 4.

    U/Iowa Geiger Tube Telescope

  5. 5.

    GSFC Cosmic Ray Telescope

  6. 6.

    UCSD Trapped Radiation Detector

  7. 7.

    UCS Ultraviolet Photometer

  8. 8.

    U/Arizona Imaging Photopolarimeter

  9. 9.

    CIT Jovian Infrared Radiometer

  10. 10.

    GE Asteroid/Meteoroid Detector

  11. 11.

    LaRC Meteoroid Detector

  12. 12.

    Flux-Gate Magnetometer (Pioneer 11 only)

The power system of the Pioneer spacecraft was designed to ensure that a successful encounter with Jupiter (with all science instruments operating) can be carried out with only three (out of four) functioning radioisotope thermoelectric generators. Instruments could be commanded on or off by ground control; late in the extended mission, when sufficient power to operate all instruments simultaneously was no longer available, a power sharing plan was implemented to ensure that the power demand on board would not exceed available power levels.

At the end of its mission, only one instrument on board Pioneer 10 remained in operation; the University of Iowa Geiger Tube Telescope. An attempt was made to power down this instrument, in order to improve the available power margin. It is not known if this command was received or executed by the spacecraft.

2.2.3 Radioisotope thermoelectric generators (RTGs)

The primary electrical source on board Pioneer 10 and 11 was a set of four radioisotope thermoelectric generators [1, 135, 350, 384, 39]. Each of these RTGs contains 18 238Pu capsules, approximately two inches (5.08 cm) in diameter and 0.2 inches (0.51 cm) thick. The total thermal power of each RTG, when freshly fueled, was ∼ 650 W. Each RTG contains two sets of 45 bimetallic thermocouples, connected in series. The thermocouples initially operated at ∼ 6% efficiency; the nominal RTG output is ∼ 4 V, 10 A. The total available power at launch on board each spacecraft was ∼ 160 W (Figure 2.14).

Excess heat from each RTG is radiated into space by a set of six heat radiating fins. The fins provide the necessary radiating area to ensure that a sufficient temperature differential is present on the thermocouples, for efficient operation.

The thermal power of the RTGs is a function of the total power produced by the 238Pu fuel, and the amount of power removed in the form of electrical energy by the thermocouples. The half-life of 238Pu is 87.74 years. The efficiency of the thermocouples decreased over the years as a result of the decreasing temperature differential between the hot and cold ends, and also as a result of aging. At the time of last transmission, each RTG on board Pioneer 10 produced less than 15 W of electrical power.

The actual shape of the RTGs is shown in Figure 2.5. It is important to note that the RTGs on board Pioneer 10 and 11 were built with the large fins that are depicted in this figure. These enlarged fins are not shown in many drawings and photographs (including Figure 2.3).

Figure 2.5
figure 5

The SNAP-19 RTGs used on Pioneer 10 and 11 (from [350]). Note the enlarged fin structure. Dimensions are in inches (1″ = 2.54 cm).

2.2.4 The electrical subsystem

Electrical power from the RTGs reached the spacecraft body via a set of ribbon cables. There, raw power from the RTGs was fed into a series of electric power supplies that produced 28 VDC for the spacecraft’s main bus, and also other voltages on secondary power buses.

Electrical power consumption on board the spacecraft was regulated to ensure a constant voltage on the various power buses on the one hand, and an optimal current draw from the RTGs on the other hand. The electrical power subsystem was designed such that the spacecraft could perform its primary mission, namely close-up observations of the planet Jupiter approximately 21 months after launch, using only three RTGs, while operating a full compliment of science instruments. Consequently, in the early years of their mission, significant amounts of excess electrical power were available on both spacecraft. The power regulation circuitry diverted this excess power to a shunt circuit, which dissipated some excess power internally, while routing the remaining excess power to an externally mounted shunt radiator.

The total power available on board at the time of launch was in excess of ∼ 160 W. This figure decreased steadily throughout the missions, obeying an approximate negative exponential law. The actual amount of power available was a function of the decay of the radioisotope fuel, the decreasing temperature differential between the hot and cold ends of thermoelectric elements, and degradation of the elements themselves. At the end of its mission, the power available on board Pioneer 10 was less than 60 W. (Indeed, a drop in the main bus voltage is the most likely reason that Pioneer 10 eventually fell silent, as the reduced voltage was no longer sufficient to operate the spacecraft’s transmitter.)

The RTGs generated electrical power at ∼ 4 VDC (Figure 2.6). Power output from each RTG was fed to a separate inverter circuit, producing 61 VAC (peak-to-peak) at ∼ 2.5 kHz. Output from the four inverters was combined and fed to the Power Control Unit (PCU), which generated the 28 VDC main bus voltage, managed the on-board battery, controlled the dissipation of excess power via a shunt circuit, and also provided power to the Central Transformer Rectifier (CTRF) component, which, in turn, supplied power at various voltages (e.g., ± 16 VDC, ± 12 VDC, +5 VDC) to other subsystems and instruments.

Figure 2.6
figure 6

Overview of the Pioneer 10 and 11 electrical subsystem (from [292]).

The power budget at any given time was a function of available power vs. spacecraft load. For instance, Figure 2.7 shows Pioneer 10’s power budget on July 25, 1981.

Figure 2.7
figure 7

Pioneer 10 power budget on July 25, 1981, taken as an example. Power readings that were obtained from spacecraft telemetry are indicated by the telemetry word in the form Cnnn. The discrepancy between generated power and power consumption is due to rounding errors and uncertainties in the nominal vs. actual power consumption of various subsystems.

The on-board battery was designed to help with transient peak loads that temporarily exceeded the capabilities of the RTGs. The battery was composed of eight silver-cadmium cells, each of which had a capacity of 5 Ah and was equipped with an individual charge/discharge bypass circuitry.

2.2.5 Propulsion and attitude control

After separation from their respective launch vehicles, the orbits of Pioneer 10 and 11 were determined by the laws of celestial mechanics. The spacecraft had only a small amount of fuel on board, used by their propulsion system designed to control the spacecraft’s spin and orientation, and execute minor course correction maneuvers.

The propulsion system consisted of three thruster cluster assemblies (see Figure 2.8), each comprising two 1 lb (∼ 4.5 N) thrusters. All thruster cluster assemblies were mounted along the rim of the HGA. One pair of clusters was oriented tangentially along the antenna perimeter, and its two thrusters were intended to be used to increase or decrease the spin rate of the spacecraft. The remaining two pairs were oriented perpendicular to the antenna plane, on opposite sides of the antenna. These two thruster cluster assemblies were used in pairs. If two thrusters pointing in the same direction were fired simultaneously, this resulted in a net change in the spacecraft’s velocity in a direction perpendicular to the antenna plane. If two clusters were fired in the opposite direction, this caused the spacecraft’s spin axis to precess. This latter type of maneuver was used, in particular, to maintain an Earth-pointing orientation of the HGA to ensure good reception of radio signals.

Figure 2.8
figure 8

An overview of the Pioneer 10 and 11 propulsion subsystem (from [292]).

Figure 2.9
figure 9

Location of thermal sensors in the instrument compartment of the Pioneer 10 and 11 spacecraft (from [292]). Platform temperature sensors are mounted at locations 1 to 6. Some locations (i.e., end of RTG booms, propellant tank interior, etc.) not shown.

The thrusters were labeled VPT (velocity and precession thruster) and SCT (spin control thruster.) VPT 1 and VPT 3 were oriented in the same direction as the HGA (the +z direction), while VPT 2 and VPT 4 were oriented in the opposite direction.

The propulsion system utilized hydrazine (N2H4) monopropellant fuel, of which ∼ 27 kg was available on board, in a 38 liter tank that was pressurized with N2. The propellant and pressurant were separated by a flexible membrane, which prevented the mixing of the liquid propellant and gaseous pressurant in the weightless environment of space. The fuel tank was located at the center of the spacecraft, and was heated by the spacecraft’s electrical equipment. Fuel lines leading to the thruster cluster assemblies were heated electrically, while the thruster cluster assemblies were equipped with small (1 W) radioisotope heating units (RHUs) containing 238Pu fuel.

The capabilities of the propulsion system are summarized in Table 2.2.

Table 2.2 Capabilities of the Pioneer 10 and 11 propulsion system.

The spacecraft’s Earth-pointing attitude was maintained as the spacecraft were spinning in the plane of the HGA, at a nominal rate of 4.8 revolutions per minute (rpm). The propulsion system had the capability to adjust the spin rate of the spacecraft, and to precess the spin axis, in order to correct for orientation errors, and to ensure that the spacecraft followed the Earth’s position in the sky as seen from on board.

The spin axis perpendicular to the plane of the HGA is one of the spacecraft’s principal axis of inertia. A wobble damper mechanism [292] dampened rotations around any axis other than this principal axis of inertia, ensuring a stable attitude even after a precession maneuver.

2.2.6 Navigation

The Pioneer 10 and 11 spacecraft relied on standard methods of deep space navigation [235, 240] (see Section 4). The spacecraft’s position was determined using the spacecraft’s radio signal and the laws of celestial mechanics. The radio signal offered a precision Doppler observable, from which the spacecraft’s velocity relative to an Earth station along the line-of-sight could be computed. Repeated observations and knowledge of the spacecraft’s prior trajectory were sufficient to obtain highly accurate solutions of the spacecraft’s orbit.

The orientation of the spacecraft was estimated from the quality of the radio communication link (i.e., the spacecraft had to be approximately Earth-pointing in order for the Earth to fall within the HGA radiation pattern.) The rate and phase of the spacecraft’s rotation was established by a redundant pair of sun sensors and a star sensor on board. These sensors (selectable by ground command) provided a roll reference pulse that was used for navigation purposes (as explained in the next paragraph) as well as by on-board science instruments. The time between two subsequent roll reference pulses was measured and telemetered to the ground.

The spacecraft also had minimal autonomous (closed loop) navigation capability, designed to make it possible for the spacecraft to restore its orientation by “homing in” on an Earth-based signal. The maneuver, called a conical scan (CONSCAN) maneuver, utilized a piston mechanism [3] with electrically heated freon gas that displaced the feed horn located at the focal point of the high-gain antenna. Unless the Earth was exactly on the HGA centerline, this introduced a sinusoidal modulation in the amplitude of the signal received from the Earth. A simple integrator circuit, utilizing this sinusoidal modulation and the roll reference pulse, triggered firings of the precession thrusters, adjusting the spacecraft’s axis of rotation until it coincided with the direction of the Earth.

Frequently, instead of CONSCAN maneuvers, “open loop” attitude correction maneuvers were used, calculated to ensure that after the maneuver, the spacecraft was oriented to “lead” the Earth, allowing the Earth to move through the antenna pattern subsequently. This reduced the frequency of attitude correction maneuvers; further, open loop maneuvers generally consumed less propellant than autonomous CONSCAN maneuvers.

Early in the mission, attitude correction maneuvers had to be executed regularly, due to the combined motion of the Earth and the spacecraft. Late in the extended mission, only two attitude correction maneuvers were needed annually, to compensate for the Earth’s motion around the Sun, and for the spacecraft’s “sideways” motion along its hyperbolic escape trajectory.

2.2.7 Communication system

The spacecraft maintained its communication link with the Earth using a set of S-band transmitters and receivers on board, in combination with three antennae.

The main communication antenna of the spacecraft was the 2.74 m diameter high-gain antenna. The antenna’s narrow beamwidth (3.3° downlink, 3.5° uplink) ensured an effective radiated power of 70 dBm, allowing communication with the spacecraft over interplanetary distances. (The original mission design anticipated signal loss some time after Jupiter encounter, but still within the orbit of Saturn; increases in the sensitivity of Earth stations allowed communication with Pioneer 10 up until 2003, when the spacecraft was over 70 AU from the Earth.)

Mounted along the centerline of the HGA was the horn of a medium-gain antenna (MGA). On the opposite (−z) side of the spacecraft, at the bottom of the main compartment was mounted a third, low-gain omnidirectional antenna (LGA). This antenna was used during the initial mission phases, before the HGA was oriented towards the Earth.

The spacecraft had two receivers and two transmitters on board, switchable by ground command. While one receiver was connected to the HGA, the other was connected to the MGA/LGA. The sensitivity of the receiver was −149 dBm; at the time of the last transmission, the spacecraft detected the Earth station’s signal at a strength of −131.7 dBm.

The spacecraft utilized two traveling wave tube (TWT) transmitters for microwave transmission. The TWTs were selectable by ground command; it was possible to power off both TWTs to conserve power (such as when CONSCAN maneuvers were performed late in the mission, when the available electrical power on board was no longer sufficient to operate a TWT transmitter and the feed movement mechanism simultaneously.)

The radio systems operated in the S-band, utilizing a frequency of ∼ 2.1 GHz for uplink, and ∼ 2.3 GHz for downlink. The transmitter frequency was synthesized on board by an independent oscillator. However, the spacecraft’s radio system could also operate in a coherent mode: in this mode, the downlink signal’s carrier frequency was phase-coherently synchronized to the uplink frequency, at the exact frequency ratio of 240/221. In this mode, the precision and stability of the downlink signal’s carrier frequency was not limited by the equipment on board. This mode allowed precision Doppler frequency measurements with millihertz accuracy.

The main function of the spacecraft’s communication system was to provide two-way data communication between the ground and the spacecraft. Data communication was performed at a rate of 16–2048 bits per second (bps). Communication from the ground consisted of commands that were decoded by the spacecraft’s radio communication subsystem. Communication to the ground consisted of measurement results from the spacecraft’s suite of science instruments, and engineering telemetry.

2.2.8 Telemetry data and its interpretation

Communication between the spacecraft and a DSN antenna took place using a variety of data formats; the format selected depended on the type of experiment that the spacecraft was ordered to perform, but typically the format gave precedence to science results over telemetry. However, telemetry information was continuously transmitted to the Earth at all times, using a small portion of the available data bandwidth.

The telemetry data stream was assembled on board by the digital telemetry unit (DTU), which comprised some ∼ 800 TTLFootnote 4 integrated circuits, and was also equipped with 49,152 bits of ferrite core data memory. A total of 10 science data formats (of which 5 were utilized) and 4 engineering data formats, yielding a total number of 18 different valid format combinations, was selectable by ground command. The DTU could operate in three modes: realtime (passing through science measurements as received from instruments), store (storing measurements in the on-board memory) and readout (transmitting measurements previously stored in on-board memory.)

When one of the engineering data formats was selected, the spacecraft transmitted only engineering telemetry. These formats were utilized, for instance, during maneuvers or spacecraft troubleshooting.

Most of the time, a science data format was used, in which case most of the bits in the telemetry data stream contained science data. A small portion, called the subcommutator, was reserved for engineering data; this part of the telemetry record cycled through all telemetry data words in sequence.

The telemetry record size was 192 bits, divided into 36 6-bit words. Depending on the telemetry format chosen, either all 36 6-bit words contained engineering telemetry, or a single engineering telemetry word was transmitted in every (or every second) telemetry record.

There were 128 distinct engineering telemetry words. Depending on the science format used, the subcommutator was present either in every transmitted record or every second record. Therefore, it may have taken as many as 256 telemetry records before a particular engineering word was transmitted. At the lowest data rate of 16 bits per second, this meant that any given parameter was telemetered to the Earth once every 51.2 minutes.

An additional 64 telemetry words were used to transmit engineering information from science instruments. These parameters were only transmitted in the subcommutator, and at the lowest available data rate, a particular science instrument telemetry word was repeated every 25.6 minutes.

The 128 engineering telemetry words were organized into 4 groups of 32 words each; it is customary to denote them using the notation Cmnn, where m = 1…4 is the group number, and nn = 01…32 is the telemetry word. Similarly, science instrument telemetry words were labeled Emnn, with m = 1…2 and nn = 01…32.

When a science or engineering telemetry word was used to convey the reading from an analog (temperature, voltage, current, pressure, etc.) sensor, the reading was digitized with 6-bit resolution [146, 402, 167], and the resulting 6-bit data word was transmitted. Associated with each telemetry word representing an analog measurement was a set of calibration coefficients that formed a 5th-order calibration polynomial. For each calibration polynomial, a range was also defined that established valid readings that could be decoded by that polynomial.

Appendix D lists selected engineering and science telemetry data words that may be relevant to the analysis of the Pioneer anomaly.

2.2.9 Thermal subsystem

The Pioneer spacecraft were equipped with a thermal control system comprising a variety of active and passive thermal control devices. The purpose of these devices was to maintain the required operating temperatures for all vital subsystems of the spacecraft.

The main spacecraft body was covered by multilayer insulating blankets [364, 365]. These blankets were designed to retain heat within the spacecraft when it was situated in deep space, far from the Sun.

To prevent overheating of the spacecraft interior near the Sun, a thermal louver system [292, 79, 299, 385] was utilized. The louvers were located at the bottom of the spacecraft, organized in a circular pattern, with additional louvers on the science compartment (Figure 2.10). These louvers were actuated by bimetallic springs that were thermally (radiatively) coupled to the main electronics platform behind the louvers. The louvers were designed to be fully open when the platform temperature exceeded 90°F, and fully close when the temperature fell below 40°F (Figure 2.11).

Figure 2.10
figure 10

The Pioneer 10 and 11 thermal control louver system, as seen from the aft (−z) direction (from [292]).

Figure 2.11
figure 11

Louver blade angle as a function of platform temperature (from [385]). Temperatures in °F ([°C] = ([°F]−32) × 5/9).

The louver system emits heat in two ways: through structural components (Figure 2.12), and through the louver blade assemblies (Figure 2.13). The total heat emitted by the louver system is the sum of the heat emitted via these two mechanisms.

Figure 2.12
figure 12

Louver structure heat loss as a function of platform temperature (from [385]). Temperatures in °F ([°C] = ([°F]−32) × 5/9).

Figure 2.13
figure 13

Louver assembly performance (from [385]). Temperatures in °F ([°C] = ([°F]−32) × 5/9).

The fuel lines extending from the spacecraft body to the thruster cluster assemblies located along the rim of the HGA were insulated by multilayer thermal blankets and were kept warm by electric heater lines. The thruster cluster assemblies contained radioisotope heater units (RHUs), designed to prevent the propellant from freezing. Additional RHUs heated the star sensor and the magnetometer assembly at the end of the magnetometer boom.

The spacecraft’s battery was mounted on the outside of the main spacecraft body, and heated by an electrical heater.

Most heat produced by electrical equipment on board was released within the insulated interior of the spacecraft. The requirement for science instruments was to leak no more than the instrument’s own power consumption plus 0.5 W to space.

Some excess electrical power was radiated away as heat by an externally mounted shunt radiator plate, which formed part of the spacecraft’s electrical power subsystem.

Most heat on board was produced by the four radioisotope thermoelectric generators. Electrical power in these generators was produced by bimetallic thermocouples that relied on the temperature difference between their hot and cold ends for power generation. Therefore, it was essential that the cold ends of the thermocouples were connected to RTG radiator fins that radiated heat into space with high efficiency.

Temperature readings from many locations throughout the spacecraft, including temperatures at six key locations on the main electronics platform, were telemetered to the ground (Figure 2.9).

With the exception of the louver system, the two adjacent hexagonal parts of the spacecraft body are covered by multilayer insulation. There is no insulation between the main and science compartments of the spacecraft body.

The surface materials and paints used to cover most major exterior surfaces are documented [292, 61, 114, 144, 206, 385]. In particular, the surfaces of the spacecraft body, HGA, and RTGs are well described, along with the thermal control louver system in terms of solar absorptance and infrared emittance (Table 2.3.). Solar absorptance is characterized by a dimensionless number (usually denoted by α) between 0 and 1 representing the efficiency with which a particular material absorbs the radiant energy of the Sun when compared to an ideal black body. Infrared emittance is similarly characterized by a dimensionless number (usually dented by ϵ) between 0 and 1 that represents the efficiency with which a material radiates heat at lower (typically, room) temperatures as compared to an ideal black body.

Table 2.3 Radiometric properties of Pioneer 10 and 11 major exterior surfaces (at launch): solar absorp-tance (α) and infrared emittance (ϵ).

The exterior surfaces of the Pioneer 10 and 11 spacecraft are covered by a variety of materials and paints. Changes in the spacecraft properties can affect/induce forces that are both of on-board and of external origin. Therefore, it is of great importance to establish the extent to which any of the spacecraft’s properties might have been changing over time.

2.3 Spacecraft operating history

The Pioneer 10 and 11 spacecraft spent about three decades in deep space. Most of what we know about the anomalous acceleration is based on data that was collected when the spacecraft were well into their second and third decades of operation. This raises an obvious question: what are the effects of aging and to what extent may aging be responsible for the anomalous acceleration?

In this section, we summarize our knowledge about the effects of aging on the spacecrafts’ subsystems.

2.3.1 Spacecraft physical configuration

The overall shape of the Pioneer 10 and 11 spacecraft is not expected to change significantly with age. The spacecraft is fundamentally a rigid body; other than the constant centrifugal force that arises as a result of the spacecraft’s rotation, there are no forces stretching, bending, or otherwise acting on the spacecraft structurally.

The Pioneer spacecraft had few moving parts. After initial boom deployment, the spacecrafts’ physical configurations remained largely unchanging, with a few notable exceptions.

Consumption of the fuel load on board resulted in small changes in the spacecrafts’ mass distribution during the large course correction maneuvers early in the Pioneer 10 and 11 missions. As the amount of fuel on board was small, and the fuel tank was situated near the spacecraft’s center-of-gravity, the effects of later attitude correction maneuvers, which consumed minuscule amounts of fuel, were likely negligible.

Some instruments had moving parts: notably, the Imaging Photopolarimeter (IPP) instrument had a telescope that was mounted on a scan platform, allowing it to be used for Jupiter imaging. Operating this instrument’s small moving parts, however, would have introduced only minute changes in the spacecraft’s mass distribution and thermal properties.

More notable was the spacecraft’s passive louver system. As discussed in Section 2.4, the louver system was designed to vent excess heat radiatively from the interior of the spacecraft. The state of the louver system can be determined as a function of the electronics platform temperatures. The position of the louver blades can significantly alter the thermal behavior of the spacecraft, by allowing a higher proportion of interior heat to escape through the louver system.

At large heliocentric distances (beyond ∼ 25 AU), the louver system is always closed, and the spacecraft’s physical configuration remains constant.

2.3.2 Changes in spacecraft mass

As we discussed in Section 2.2.1, the nominal launch mass of the Pioneer 10 and 11 spacecraft was ∼ 260 kg, of which ∼ 30 kg was propellant and pressurant.Footnote 5 The spacecraft mass slowly decreased, primarily as a result of propellant usage. Additionally, small mass losses may occur due to fuel leaks, pressurant outgassing, He outgassing (α particles) from the RTGs and RHUs, and possibly, outgassing from the spacecraft batteries.

Pioneer 10 used only a moderate amount of propellant, as it performed no major trajectory correction maneuvers. If the propellant used amounted to one quarter of the propellant on board, this means that Pioneer 10’s mass would have decreased to ∼ 250 kg late in its mission (the 2002 JPL study used the nominal value of 251.883 kg). Further, it should be noted that most propellant usage occurred prior to Jupiter encounter; afterwards, Pioneer 10 only used minimal amounts of propellant for precession maneuvers, needed to keep its antenna aimed at the Earth.

Pioneer 11 performed major trajectory correction maneuvers en route to Jupiter and Saturn. The maneuvers were in order to allow it to follow a precisely calculated orbit that utilized a gravitational assist from Jupiter that was needed to set up the spacecraft for its encounter with Saturn. As a result, Pioneer 11 is believed to have used significantly more propellant than its twin, perhaps three quarters of the total available on board. The spacecraft’s mass, therefore, may have decreased to ∼ 232 kg following its encounter with Saturn (the mass used in the 2002 JPL study was somewhat higher, 239.73 kg [27]).

We note that these figures are crude estimates, as the actual amount of propellant on board is not telemetered. Sensors inside the propellant tank did offer temperature and pressure telemetry, but these sensors were not sufficiently reliable for a precise estimate of the remaining fuel on board.

The spacecraft can also lose mass due to outgassing. Two possible sources of outgassing are helium outgassing from the radioactive fuel on board in the radioisotope thermoelectric generators and radioisotope heater units, and chemical outgassing from the spacecraft’s battery. An upper limit of 18 g on helium outgassing can be established using the known physical properties of the 238Pu fuel (see Section 4), which is not significant. Similarly, the amount of gas that can escape from the spacecraft’s batteries is small, especially in view of the fact that the batteries performed nominally far longer than anticipated: under no circumstances can it exceed the battery mass, but in all likelihood, and especially in view of the fact that the batteries performed nominally throughout the mission, any outgassing is necessarily limited to a very small fraction of the ∼ 2.35 kg battery mass (see Section 4). Therefore, outgassing cannot have played a major role in the evolution of the spacecraft mass; furthermore, any mass loss due to outgassing is dwarfed by uncertainties in the mass of the remaining fuel inventory.

Could the spacecraft have gained mass, for instance, by collecting dust particles from the interplanetary medium? In situ measurements by the Pioneer spacecraft themselves provide an upper limit on the amount of dust encountered by the spacecraft. After passing Jupiter’s orbit, the dust flux measured by Pioneer 10 remained approximately constant, at 3 × 10−6 m−2s−1 particles [178]. The upper limit on particle sizes is 10−4 kg. Even assuming that all particles had masses near this upper limit, the total amount of mass that could have accumulated on the spacecraft over the course of 20 years would be no more than ∼ 1 kg. However, given that the spacecraft is moving through interplanetary space at a velocity of 10 km/s or higher, these assumptions on particle size would imply a dust density of ∼ 10−14 kg/m3, which is many orders of magnitude higher than more realistic estimates (see Section 4.3.5). Therefore, the actual dust mass accumulated on the spacecraft cannot be more than a few grams at the most; consequently, this mechanism for mass increase can also be safely ignored.

2.3.3 Instrumentation

It is not known how on-board instrumentation (i.e., telemetry sensors) respond to aging. What we know is that many sensors stopped providing usable readings when measured values (e.g., temperatures) dropped outside calibrated ranges [378, 379, 395, 397]. Other sensors continued to provide consistent readings, with no indication of sensor failure.

However, there were a few sensor anomalies that may be due to age-related sensor defects. Most notable among these are the anomalous readings from the propellant tank of Pioneer 10 (described in Section 2.3.6).

It is also unknown how on-board instrumentation responds when their supply voltage drops below the nominal level. Late in its mission, the electrical power subsystem on board Pioneer 10 no longer had sufficient power to maintain the nominal main bus voltage of 28 VDC. As this coincides with changes in physical sensor readings (e.g., drops in temperature), the extent to which those readings are affected by the drop in voltage is not readily evident.

2.3.4 Electrical system

Figure 2.14
figure 14

Changes in total RTG electrical output (in W) on board Pioneer 10 (left) and 11 (right), as computed using the missions’ on-board telemetry.

Insofar as we can determine from telemetry, the electrical subsystems on board Pioneer 10 and 11 performed nominally throughout the missions, so long as sufficient electrical power was available from the RTGs. The on-board chemical batteries remained functional for many years; eventually, due to irreversible chemical changes and decreasing temperatures, the batteries ceased functioning.

2.3.5 Radioisotope thermoelectrical generators

The effects of aging on the RTGs are complex. Internally, aging causes a degradation of the bimetallic thermocouples, contributing to their loss of efficiency and the decrease in RTG electrical power output. Externally, it has been conjectured [327] that the RTG exterior surfaces may have aged due to solar bleaching and impact by dust particles. The extent to which such degradation may have occurred (if at all) is unknown. The resulting fore-aft asymmetry may be a significant source of unaccounted-for acceleration in the approximately sunward direction.

2.3.6 Propulsion system

The propellant pressure sensor on board Pioneer 10 began to show anomalous behavior in June 1989. Propellant pressure, which remained steady up to this point, only decreasing slowly as a result of cooling and occasional propellant usage, suddenly began to show a sharp decay, dropping from over 300 psia down to about 150 psia by January 1992 (Figure 2.15). At this time, the propellant pressure instantaneously increased to its pre-1989 value of ∼ 310 psia, after which it began dropping again. There were no corresponding changes or other anomalies in the observed propellant temperature and expellant temperature. Therefore, the most likely explanation for this anomaly is a sensor malfunction, not a real loss of fuel or propellant.

Figure 2.15
figure 15

Propulsion tank pressure (in pounds per square inch absolute; 1 psia = 6.895 kPa) on board Pioneer 10. The three intervals studied in [27] are marked by roman numerals and separated by vertical lines.

On December 18, 1975, subsequent to an attempted maneuver, and as a result of a stuck thruster valve, the spin of Pioneer 11 increased dramatically, from a spin rate of ∼ 5.5 revolutions per minute (rpm) to ∼ 7.7 rpm (Figure 2.16, right panel). Fortunately, the thruster ceased firing before the spin rate increased to a value that would have threatened the spacecraft’s structural integrity or compromised its ability to carry out its mission.

Figure 2.16
figure 16

On-board spin rate measurements (in rpm) for Pioneer 10 (left) and Pioneer 11 (right). The sun sensor used on Pioneer 10 for spin determination was temporarily disabled between November 1983 and July 1985, and was turned off in May 1986, resulting in a ‘frozen’ value being telemetered that no longer reflected the actual spin rate of the spacecraft. Continuing spot measurements of the spin rate were made using the Imaging Photo-Polarimeter (IPP) until 1993. The anomalous increase in Pioneer 11’s spin rate early in the mission was due to a failed spin thruster. Continuing increases in the spin rate were due to maneuvers; when the spacecraft was undisturbed, its spin rate slowly decreased, as seen in Figure 2.17.

2.3.7 Attitude control and spin

Encounter with the intense radiation environment in the vicinity of Jupiter damaged the star sensor on board Pioneer 10. As the sun sensors operate only up to a distance of ∼ 30 AU, Pioneer 10 had operated without a primary roll reference for many years before the end of its mission. (The rate of Pioneer 10’s spin was determined from measurements taken by its Imaging Photo-Polarimeter (IPP) instrument and other methods [349].)

The nominal spin rate of the Pioneer 10 and 11 spacecraft was 4.8 rpm. This spin rate was achieved by reducing the spacecraft’s initial rate of spin, provided by the launch vehicle, in successive stages, first by firing spin/despin thrusters, and then by extending the RTG and magnetometer booms. Later, spin could be precisely adjusted and corrected by the spin/despin thrusters.

On Pioneer 11, due to the spin thruster anomaly described in the previous section, the spacecraft’s spin remained at an abnormally high value. In fact, it further increased, presumably as a result of fuel leaks, all the way up to ∼ 8.4 rpm at the time of the last available telemetry data point, on February 11, 1994.

Figure 2.17
figure 17

Zoomed plots of the spin rate of Pioneer 11. On the left, the interval examined in [27] is shown; maneuvers are clearly visible, resulting in discrete jumps in the spin rate. The figure on the right focuses on the first half of 1987; the decrease in the spin rate when the spacecraft was undisturbed is clearly evident.

Meanwhile, Pioneer 10’s spin slowly decreased over time, probably due to a combination of effects that may include fuel leaks as well as the thermal recoil force and associated change in angular momentum.

The spin rates of Pioneer 10 and 11 are shown in Figure 2.16. Spin was measured on board using one of several sensors, namely a star sensor and two sun sensors. The purpose of these sensors was to provide a roll reference pulse that could then be used to synchronize other equipment, including the IPP instrument and the navigational system.

The sun sensors required a minimum angle between the spacecraft’s spin axis and the spacecraft-Sun line. Further, they required that the spacecraft be within a certain distance of the Sun, in order for a reliable roll reference pulse to be generated. For these reasons, the sun sensors could not be used to provide a roll reference pulse once the spacecraft were more than ∼ 30 AU from the Sun.

There were no such limitations on the star sensor; however, the star sensor on board Pioneer 10 ceased functioning shortly after Jupiter encounter, probably due to radiation damage suffered while the spacecraft traversed the intense radiation environment in the gas giant’s vicinity.

As a result, Pioneer 10 lost its roll reference source when its distance from the Sun increased beyond ∼ 30 AU. Although the roll reference assembly continued to provide roll reference pulses at the last “frozen in” rate, this rate no longer matched the actual rate of revolution of the spacecraft.

Nevertheless, it was important to know the spin rate of the spacecraft with reasonable precision, in order to be able to carry out precession maneuvers reliably, and also because the spacecraft’s spin affected the spacecraft’s radio signal and the Doppler observable. For this reason, the IPP was reused as a surrogate star sensor, its images of the star field providing a reference that could then be used by the navigation team to compute the actual spin rate of the spacecraft on the Earth [349]. It was during the time when the IPP instrument was used to compute the rate of spin that a spin anomaly was detected. The spin-down rate of Pioneer 10 suddenly grew in 1990, and then eventually returned to its approximate previous value (Figure 2.16, left panel.)

Very late in Pioneer 10’s mission, when the power on board was no longer sufficient to operate the IPP instrument, crude estimates of the spin rate were made using navigational data.

In contrast to the spin behavior of Pioneer 10, the spin rate of Pioneer 11 continued to increase after the initial jump following the thruster anomaly. However, a close look at detailed plots of the spin rate reveal a more intricate picture. It seems that Pioneer 11’s spin rate was actually decreasing between maneuvers, when the spacecraft was undisturbed; however, each precession maneuver increased the spacecraft’s spin rate by a notable amount. The rate of decrease between successive maneuvers was not constant, suggesting that fuel leaks played a more significant role in Pioneer 11’s spin behavior than in Pioneer 10’s.

2.4 Thermal control subsystem

A quick look at the Pioneer spacecraft (Figure 2.3) is sufficient to see that the spacecraft’s thermal radiation pattern may be anisotropic: whereas one side is dominated by the high-gain antenna, the other side contains a thermal louver system designed to vent excess heat into space. The RTGs are also positioned slightly behind the HGA, making it likely that at least some of their heat is reflected in the −z direction.

This leads to the conclusion that anisotropically rejected thermal radiation cannot be ignored when we evaluate the evolution of the Pioneer 10 and 11 trajectories, and must be accounted for with as much precision as possible.

Far from the Sun and planets, the only notable heat sources on board Pioneer 10 and 11 are internal to the spacecraft. We enumerate the following heat sources:

  • Waste heat from the radioisotope thermoelectric generators;

  • Heat produced by electrical equipment on board;

  • Heat from small radioisotope heater units;

  • Transient heating from the propulsion system.

While strictly speaking it is not a heat source, one must also consider microwave radiation from the spacecraft’s radio transmitter and HGA, as this radiation also removes what would otherwise appear as thermal energy from within the spacecraft.

2.4.1 Waste heat from the RTGs

The RTGs are the most substantial sources of heat on board. Each of the four generators on board the spacecraft produced ∼ 650 W of power at the time of launch, of which ∼ 40 W was converted into electrical energy; the rest was radiated into space as waste heat. The radiation pattern of the RTGs is determined by the shape and composition of their radiating fins (see Figure 2.5), but, notwithstanding the possible effects of aging, it is believed to be fore-aft symmetrical.

The amount of power generated by the RTGs is determined by the radioactive decay of the 238Pu fuel on board. The radioactive half-life of the fuel is precisely known (∼ 87.74 years), and the total power output of each RTG was measured before launch (Table 2.4.).

Table 2.4 RTG total power measurements prior to launch [350]. The reported accuracy of these measurements is 0.1 W, but values are rounded to the nearest W. RTG numbering corresponds to the actual number of units built.

The amount of power removed from each RTG in the form of electrical energy is telemetered to the ground. (Specifically, the RTG output voltage and current for each individual RTG is telemetered.) The remainder of the RTG power is radiated in the form of waste heat.

For each RTG, two temperature measurements are also available. One sensor, internal to the RTG, measures the temperature at the hot end of the bimetallic thermocouples. The other sensor measures the temperature near the root of one of the RTG radiating fins.

2.4.2 Electrical heat

Next to the RTGs, the second most significant source of thermal radiation on board is the set of electrical equipment operating on the spacecraft. (Figure 2.4 shows the internal arrangement of components.) From a thermal perspective, nearly all electrical systems on board perform only one function: they convert electrical energy into waste heat. The one exception is the spacecraft’s radio transmitter, converting some electrical energy into a directed beam of radio frequency energy.

Early in the missions, the RTGs produced more electrical power than what was needed on board. The main bus voltage on board was maintained at a constant value by a power supply circuit, while excess electrical power was converted into heat. The shunt regulator controlled a variable current, which flowed through the regulator itself and an external radiator plate. The external radiator plate acted as an ohmic resistor, and the power radiated by it can be easily computed from the shunt current telemetry (Figure 2.7). The shunt radiator is mounted such that its thermal radiation is primarily in a direction that is perpendicular to the spacecraft spin axis.

Some electrical heat is generated outside the spacecraft body: notable items include electrical heaters for the fuel lines that deliver fuel to the thruster cluster assemblies along the antenna rim, an electrical heater for the spacecraft battery, and some scientific instruments.

Most of the electrical power not radiated away by the spacecraft antenna or the shunt radiator is converted into heat inside the spacecraft body. As the spacecraft is very close to a thermal steady state, all the electrical heat produced internally must be radiated into space. This thermal radiation is likely to cause a measurable sunward acceleration, for several reasons. First, the spacecraft body is heavily insulated by multilayer thermal insulation; however, at the bottom of the spacecraft body is the thermal louver system designed to vent excess heat. Even in its closed state, the effective emissivity of this louver system is much higher than that of the multilayer insulation, so this is the preferred direction in which heat escapes the spacecraft. Second, the spacecraft body is situated behind the high-gain antenna; even if the thermal radiation pattern of the spacecraft body were isotropic, the back of the HGA would preferentially reflect a significant portion of this heat in the aft direction.

Figure 2.7 also indicates how the power consumption by various pieces of equipment on board can be computed from telemetry.

The thermal state of the interior of the spacecraft is characterized in a redundant manner. In addition to the thermal power of various components that can be computed from telemetry, there exist numerous temperature sensors on board, most notably among them six platform temperature sensors, which are the most likely to measure ambient temperatures (as opposed to sensors that, say, are designed to measure the temperature of a specific electrical component, such as a transistor amplifier.)

Much less is known about heat escaping the interior of the spacecraft through other routes. The spacecraft’s science instruments utilize various holes and openings in the spacecraft body in order to collect information from the environment. Design documentation prescribes that no instrument can lose more heat through the opening than its own power consumption plus 0.5 W; however, the actually heat loss per instrument is not known.

2.4.3 Radioisotope heater units

A further source of continuous heat on board is the set of radioisotope heater units (RHUs) placed at strategic locations to maintain operating temperatures. These RHUs are capsules containing a small amount of 238Pu fuel, generating ∼ 1 W of heat (Figure 2.18).

Figure 2.18
figure 18

1 W radioisotope heater unit (RHU). From [292].

Each thruster cluster assembly housed three RHUs that prevented the freezing of thruster valves. One RHU was located at the sun sensor, while an additional RHU was placed at the magnetometer. (Some reports suggest that a 12th RHU may also have been on board one or both spacecraft.)

The thruster cluster assemblies were designed to radiate heat in a direction perpendicular to the spin axisFootnote 6. There are no known asymmetries of any thermal radiation from the magnetometer assembly at the end of the magnetometer boom. Therefore, it is unlikely that thermal radiation from the RHUs in general contributed much thermal recoil force in the fore-aft direction.

2.4.4 Waste heat from the propulsion system

The spacecraft’s propulsion system, when operating, produced significant amounts of heat; as the hydrazine monopropellant underwent a chemical reaction in the presence of a catalyst in the spacecraft’s thrusters, the thrusters and thruster cluster assemblies warmed up to several hundred degrees Centigrade. This heat was radiated into space as the assemblies cooled down to their pre-maneuver temperatures after a thruster firing event over the course of several hours.

However, the uncertainty in the magnitude of velocity change caused by the thruster event itself dwarfs any acceleration produced by the radiation of this residual heat. For this reason, heat from the propulsion system does not need to be considered when accounting for thermal recoil forces.

2.4.5 The energy radiated in the radio beam

To complete our discussion of thermal radiation emitted by the spacecraft, we must also consider the spacecraft’s radio beam, for two reasons. First, any energy radiated by the radio beam is energy that is not converted into heat inside the spacecraft body. Second, the radio beam itself produces a recoil force that is similar in nature to the thermal recoil force.

The nominal power of the radio transmitter is 8 W, and the radio beam is highly collimated by the high gain antenna. Nevertheless, some loss occurs around the antenna fringes, and the beam itself also has a spread. Assuming that ∼ 10% of the radio beam emitted by the feedhorn misses the antenna dish, at an approximate angle of 45°, we can calculate an efficiency of ∼ 0.83 at which the antenna converts the emitted radio energy into momentum [327].

The actual power of the radio beam may not have been exactly 8 W. The output of the traveling wave tube oscillator that generated this microwave energy was measured on board and telemetered to the ground (Figure 2.19). Especially in the case of Pioneer 10, we note the variability of the TWT power near the end of mission. This corresponds to the drop in main bus voltage when the power available on board was no longer sufficient to maintain nominal voltages. It is unclear, therefore, if the measured drop in output power is an actual drop or a sensor artifact.

Figure 2.19
figure 19

The emitted power (measured in dBm, converted to W) of the traveling wave tube transmitter throughout the mission, as measured by on-board telemetry. Left: Pioneer 10, which used TWT A (telemetry word C231). Right: Pioneer 11, initially using TWT A but switching to TWT B (telemetry word C214) early in its mission.

2.4.6 Thermal measurements in the telemetry

The thermal history of Pioneer 10 and 11 can be characterized accurately, and in detail, with the help of the recently recovered telemetry files and project documentation.

The telemetry data offer a redundant picture of the spacecrafts’ thermal state. On the one hand, the amount of power available on board can be computed from electrical readings. On the other hand, a number of temperature sensors on board offer a coarse temperature map of the spacecraft.

The electrical state of Pioneer 10, when the spacecraft was at 25 AU from the Sun, is shown in Figure 2.7. While this figure represents a snapshot of the spacecraft’s state at a particular moment in time, this information is available for the entire duration of both Pioneer missions.

The thermal state of the spacecraft body can be verified using the readings of six temperature sensors that were located at various points on the electronics platform: four in the main compartment, two in the adjacent compartment that housed science instruments (see locations 1–6 in Figure 2.9). The temporal evolution of these temperature readings agrees with expectations: outlying sensors show consistently lower temperatures, and all temperatures are dropping steadily, as the distance between the spacecraft and the Sun increases while the amount of power available on board decreases (Figures 2.20 and 2.21).

Figure 2.20
figure 20

Platform temperatures (left) and RTG fin root temperatures (right) on board Pioneer 10. Temperatures in °F ([° C] = ([°F]−32) × 5/9).

Figure 2.21
figure 21

Platform temperatures (left) and RTG fin root temperatures (right) on board Pioneer 11. Temperatures in °F ([° C] = ([°F]−32) × 5/9).

There was a large number of additional temperature sensors on board (see Appendix D). However, these temperature sensors measured the internal temperatures of on-board equipment and science instruments. One example is the hot junction temperature sensor inside the RTGs, measuring the thermocouple temperature at its hot end. These readings may be of limited use when assessing the overall thermal state of the spacecraft. Nevertheless, they are also available in the telemetry.

The thermal behavior of the spacecraft was an important concern when Pioneer 10 and 11 were designed. Going beyond a general description [292], much of the work that was done to ensure proper thermal behavior is documented in a review document of the Pioneer 10 and 11 thermal control subsystem [385]. Specifically, this document provides detailed information about the thermal properties of components internal to the spacecraft, and about the anticipated maximum heat losses through various spacecraft structural and other components.

Detailed information about the SNAP-19 RTGs used on board Pioneer 10 and 11 is also available [350].

2.4.7 Surface radiometric properties and thermal behavior

It has been suggested (see, e.g., [327]) that the material properties of these surfaces may have changed over time, introducing a time-dependent anisotropy in the spacecraft’s thermal properties. For instance, solar bleaching may affect spacecraft surfaces facing the Sun [61], while surfaces facing the direction of motion may be affected by the impact of interplanetary dust particles.

Note that through most of their operating lives, the high-gain antennas of the Pioneer 10 and 11 spacecraft were always pointing in the approximate direction of the Sun. Therefore, the same side of the spacecraft: notably, the interior surfaces of their HGAs and one side of each RTG, were exposed to the effects of solar light, including ultraviolet radiation, and charged particles. It is not unreasonable to assume that this continuous exposure may have altered the visible light and infrared optical properties of these surfaces, introducing in particular, a fore-aft asymmetry in the thermal radiation pattern of the RTGs. On the other hand, any such effects would be mitigated by the fact that the spacecraft were receding from the Sun very rapidly, and spent most of their operating lives at several AUs or more from the Sun, receiving only a fraction of the solar radiation that is received, for instance, by Earth-orbiting satellites of comparable age.

Another possible effect may have altered the infrared radiometric properties of spacecraft surfaces facing in the direction of motion (which, most of the time, would be surfaces facing away from the Sun.) As the spacecraft travels through the interplanetary medium at speeds in excess of 12 km/s, impact by charged particles and dust may have corroded these surfaces. Once again, this may have introduced a fore-aft anisotropy in the infrared radiometric properties of the spacecraft, most notably their RTGs.

NASA conducted several experiments to investigate the effect of space exposure on the thermal and optical properties of various materials. One investigation, called the Thermal Control Surfaces (TCS) experiment [411], examined the long-term effects of exposure of different materials placed on an external palette on the International Space Station. Although results of this test are quite important, the near-Earth environment is quite different from that of deep space. Concerning solar bleaching, although the spacecraft spent decades in space, most of the time was spent far away from the Sun, resulting in a low number of “equivalent Sun hours” (ESH). The cumulative exposure to solar radiation of the Sun-facing surfaces of Pioneer 10 and 11 is less than the amount of solar radiation test surfaces were exposed to during the TCS experiment, in which test surfaces were also exposed to the atomic oxygen of the upper atmosphere, which is not a consideration in the case of the Pioneer spacecraft.

The most pronounced effect on coatings in the deep space environment is due to exposure to solar ultraviolet radiation [61]. The ESH for the Pioneer 10 and 11 spacecraft is approximately 3000 hours, most of which (> 95%) were accumulated when the spacecraft were relatively close (< 15 AU) to the Sun. The exterior surfaces of the RTGs were covered by a zirconium coating in a sodium silicate binder [350]. For similar inorganic coatings, the most pronounced effect of prolonged solar exposure is an increase (Δα ∼ 0.05) in solar absorptance. No data suggest a noticeable change in infrared emittance.

The effects of exposure to dust and micrometeoroid impacts in the interplanetary environment do not result in significant optical damage, defined in terms of changes in solar absorptance or infrared emittance [61]. This is confirmed by the Voyager 1 and 2 spacecraft that had unprotected camera lenses that were facing the direction of motion, yet suffered no observable optical degradation. This fact suggests that the optical effects of exposure to the interplanetary medium on the spacecraft are negligible (see discussion in [27, 392]).

3 Pioneer Data Acquisition and Preparation

Discussions of radio science experiments with spacecraft in the solar system require a general knowledge of the sophisticated experimental techniques used by NASA’s Deep Space NetworkFootnote 7 (DSN). Specifically, for the purposes of the Pioneer Doppler data analysis one needs a general knowledge of the methods and techniques implemented in the radio science subsystem of the DSN. One also needs an understanding of how spacecraft telemetry is collected at the DSN, distributed and used to assess the state of the spacecraft systems at a particular epoch.

Since its beginnings in 1958, the DSN underwent a number of major upgrades and additions. This was necessitated by the needs of particular space missionsFootnote 8. The history of the Pioneer 10 and 11 projects is inextricably connected to that of DSN. Due to the continuing improvements of the entire network, Pioneer 10 was able to communicate with the project team for over 30 years — far beyond the originally planned operational life of 3 years or less.

In this section, we discuss the history of the DSN, its current status, and describe the DSN antennas and operations in support of deep space missions. We also review the methods and techniques implemented in the radio science subsystem of the DSN that is used to obtain the radio tracking data, from which, after analysis, results are generated.

3.1 The Deep Space Network

Conceived at the dawn of the space era in the late 1950s, the DSN is a collection of radio tracking stations positioned around the globe [128, 152, 374], along with ground data transfer and data processing systems, designed to maintain continuous two-way communication, including commands and telemetry, with a variety of deep space vehicles located throughout the solar system.

The first DSN complex was constructed near the Jet Propulsion Laboratory (JPL), in Goldstone, California. It was here that the first large (26 m) DSN antenna, DSS-11 “Pioneer” (named after, unsurprisingly, the early Pioneer program) was constructed.

The first overseas DSN location was in Woomera, Australia, about 350 km north of Adelaide. DSS-41, another 26 m antenna, was constructed there in 1960. The second overseas DSN complex was built near Johannesburg, South Africa. The 26 m antenna of DSS-26 was constructed there in 1961. Together, Goldstone, Woomera and Johannesburg made it possible to initiate and maintain continuous communication with distant spacecraft at any time of the dayFootnote 9, marking the beginning of the DSN.

Today, three locations — Goldstone, California; Madrid, Spain; and Canberra, Australia — form the backbone of the DSN. Presently, each of the three DSN complexes hosts several tracking stations with different capabilities and antennas of different sizes [152, 241].

3.1.1 DSN tracking stations

The primary purpose of the DSN is to maintain two-way communication with distant spacecraft. The DSN can send, or uplink, command instructions and data, and it can receive, or downlink, engineering telemetry and scientific observations from instruments on board these spacecraft.

The DSN can also be used for precision radio science measurements, including measurements of a signal’s frequency and timing observations. These capabilities allow the DSN to perform, for instance, Doppler measurements of line-of-sight velocity, ranging measurements to suitably equipped spacecraft, occultation experiments when a spacecraft flies behind a planetary body, and planetary radar observations [38].

The antennas of the DSN are capable of bidirectional communication using a variety of frequencies in the L-band (1–2 GHz), S-band (2–4 GHz), X-band (8–12 GHz) and, more recently, K-band (12–40 GHz). Early spacecraft used the L-band for communication [241] but within a few years, S-band replaced L-band as the preferred frequency band. The Pioneer 10 and 11 spacecraft used S-band transmitters and receivers. X-band and, more recently, K-band is used on more modern spacecraft.

In addition to the permanent DSN complexes that presently exist at Goldstone, California, Madrid, Spain, and Canberra, Australia, and the now defunct complexes in Woomera, Australia and Johannesburg, South Africa, occasionally, non-DSN facilities (e.g., the Parkes radio observatory in Australia) were also utilized for communication and navigation. During the long lifetime of the Pioneer project, nearly all the large antennas of DSN tracking stations and also some non-DSN facilities participated in the tracking of the Pioneer 10 and 11 spacecraft at one time or another.

The capabilities of the DSN evolved over the years. By the time of the launch of Pioneer 10, the DSN was a mature network comprising a number of 26 m tracking stations at four locations around the globe, a new 64 m tracking station in operation at Goldstone, California, and two more 64 m tracking stations under construction in Australia and Spain. The Goldstone facility was connected to the then new Space Flight Operations Facility (SFOF) built at the JPL via a pair of 16.2 kbps communication links (Figure 3.1), allowing for the real-time monitoring of spacecraft telemetry [241].

Figure 3.1
figure 22

DSN facilities planned to be used by the Pioneer project in 1972 [334]. DSIF stands for Deep Space Instrumentation Facility, GCF is the abbreviation for Ground Communications Facility, while SFOF stands for the Space Flight Operations Facility.

The most important characteristics of a DSN tracking station can be described using parameters such as antenna size, antenna (mechanical) stability, receiver sensitivity, and oscillator stability. These characteristics determine the accuracy with which the DSN can perform radio science investigations and, in particular with respect to the Pioneer 10 and 11 Doppler frequency measurements.

3.1.2 Antennas of the DSN

Large, precision-steerable parabolic dish antennas are the most recognizable feature of a DSN tracking station. In 1972, several 26 m antennas were in existence at the Goldstone, Madrid, Johannesburg and Woomera facilities. Additionally, the 64 m antenna at Goldstone was already operational, while two 64 m antennas at Madrid and Woomera were under construction (Table 3.1.).

Table 3.1 Pre-launch planned use of Deep Space Instrumentation Facilities in support of the Pioneer 10 and 11 projects (from [334]). Some stations only became operational in 1973. Several stations were decommissioned during the lifetime of the Pioneer project (DSS-11 and DSS-62 in 1981, DSS-12 in 1996, and DSS-42 and DSS-61 in 1999).

To appreciate the impressive performance of the DSN in support of the Pioneer 10 and 11 missions in deep space, one needs to be able to evaluate the factors that contribute to the sensitivity of an antenna. The maximum strength of a signal received by a tracking station or spacecraft is a function of antenna area and distance between the transmitting and receiving stations. The gain G of a parabolic antenna of diameter d at wavelength λ is calculated as

$$G = {\pi ^2}{{{d^2}} \over {{\lambda ^2}}} = 4\pi {A \over {{\lambda ^2}}},$$

where A = πd2/4 is the antenna area. Space loss due to the distance R between a transmitting and a receiving antenna is calculated, in turn, as

$${L_{{\rm{spc}}}} = {{{\lambda ^2}} \over {{{(4\pi R)}^2}}}.$$

The thermal noise power N (also known as the Johnson-Nyquist noise) of a receiver is calculated from the receiver’s system noise temperature Ts using

$$N = kB{T_s},$$

where B is the bandwidth (Hz) and k = 1.381 × 10−23 JK−1 is Boltzmann’s constant. The S-band system noise temperature of a 64 m antenna (DSS-14) was 28 K in 1971 [335]. Later, with the installation of new receivers, for an antenna at 60° elevation, the system noise was reduced to 12.9 K [307]. The receiver bandwidth in 1971 was 12 Hz, reduced to 3 Hz for the Block IV receivers and then eventually to as low as 0.1 Hz for the Block V receivers (details are in Section 3.1.3).

Figure 3.2
figure 23

DSN performance estimate throughout the primary missions of Pioneer 10 and 11. Adapted from [339].

Antenna sensitivity is measured by its signal-to-noise ratio, relating the power S of the received signal to the noise power N of the receiver. The strength of the received signal can be calculated as [152]:

$${S \over N} = {G_T}{G_R}{L_{{\rm{spc}}}}{{{P_T}} \over N} = {{{P_T}{A_T}{A_R}} \over {{\lambda ^2}{R^2}N}},$$

where PT is the transmitter power, AT is the effective area of the transmitting antenna, and AR is the effective area of the receiving antenna.

To calculate the actual signal-to-noise ratio, one must also take into account additional losses The effective area of an antenna may be less than the area of the dish proper, for instance due to obstructions in front of the antenna surface (e.g., struts, assemblies, other structural elements) For a 64 m DSN antenna, these losses amount to 2.7 dB [335], whereas for the 2.74 m parabolic dish antenna of Pioneer 10 and 11, these losses are about 3.7 dB.

Further (circuit, modulation, pointing) losses must also be considered. For the downlink from Pioneer 10 and 11 to the ground, the sum total of these losses is about 10.4 dB.

The signal-to-noise ratio also determines the bit error rate at various bit rates through the equation

$${S \over N} = {{{E_b}} \over {{N_0}}}{{{f_b}} \over B},$$

where Eb, is energy per bit, N0 is the noise energy per Hz, and fb is the bit rate. From this and Equation (3.3), the bit error rate pe can be computed as [374]:

$${p_e} = 0.5\,\,{\rm{erfc}}\left({\sqrt {{{{E_b}} \over {{N_0}}}}} \right) = 0.5\,\,{\rm{erfc}}\left({\sqrt {{S \over {k{f_b}{T_s}}}}} \right).$$

The discussion above allows one to present a typical downlink communications power budget for Pioneer 10, which is given in Table 3.2 (adapted from [336]).

Table 3.2 Pioneer 10 Jupiter downlink carrier power budget for tracking system [336].

Using the facilities in existence in 1973, the DSN would have been able to track Pioneer 10 and 11 up to a geocentric distance of ∼ 22 AU, but not beyond. However, due to numerous improvements of the DSN it was in fact possible to track Pioneer 10 all the way to over ∼ 83 AU from the Earth. Not the least of these improvements was an increase in antenna size, when the DSN’s 64 m antennas were enlarged to 70 m, and many of the 26 m antennas (notably, DSS-12, DSS-42, and DSS-61 from Table 3.1) were enlarged to 34 m.

In addition to the stations used initially (see Table 3.1) for communication with Pioneer 10 and 11, over the years many other stations were utilized, which are listed in Table 3.3.

Table 3.3 Additional DSN stations in operation during the Pioneer mission lifetime that may have been used for tracking Pioneer 10 and 11.

In order to perform precision tracking, the location of these radio stations must be known to high accuracy in the same coordinate frame (e.g., a solar system barycentric frame) in which spacecraft orbits are calculated.

This task is accomplished in two stages. First, station locations are given relative to a a geocentric coordinate system, such as the International Terrestrial Reference Frame (ITRF). Second, a conversion from the geocentric coordinate system to the appropriate solar system barycentric reference frame is performed.

For operating DSN stations, station location information is readily available, e.g., from NASA’s Navigation and Ancillary Information Facility (NAIFFootnote 10). Such information usually consists of station coordinates at a given epoch, and station drift (e.g., as a result of continental drift.) Higher precision station data (e.g., taking into account the effects of tide) is also available, but such precision is not required for the tracking of Pioneer 10 and 11.

Station data is harder to come by for stations that have been decommissioned or moved. Some decommissioned stations are listed in Table 3.4Footnote 11.

Table 3.4 Station location information for a set of decommissioned DSN stations. ITRF93 Cartesian coordinates (in km) and velocities (north, east, vertical, in cm/year) are shown. For DSS-12 and DSS-61, station information is for dates prior to July 1, 1978 and August 9, 1979, respectively.

3.1.3 DSN receivers

Antenna size may have been the visually most apparent change at a DSN complex, but it was upgrades of DSN receiver hardware that resulted in a really significant improvement in a tracking station’s capabilities.

Receivers of the DSN have been improved on a continuous basis during the lifetime of Pioneer 10 and 11. These improvements were a result of other mission requirements, but the Pioneer project benefited: far beyond the predicted range of 22 AU, it was possible to maintain communication with Pioneer 10 when it was at an incredible 83 AU from the Sun. (The same DSN complex is now used to communicate with Voyager I spacecraft from heliocentric distances of over 110 AU.)

The receiver can contribute to a tracking station’s sensitivity in three different ways. First, the bandwidth of the receiver (loop bandwidth) can be reduced, increasing the signal-to-noise ratio. Two additional improvements, lowering the system noise temperature and eliminating other sources of signal attenuation, not only increase the signal-to-noise ratio but also decrease the bit error rate.

The “Block III” DSN receivers in use at the time of the launch of Pioneer 10 and 11 had an S-band system noise temperature of 28 K, and a receiver loop bandwidth of 12 Hz. These receivers were replaced in 1983–1985 by “Block IV” receivers in which the S-band system noise was reduced to 14.5 K in receive only mode, and the receiver loop bandwidth was reduced to 3 Hz. Further improvements came with the all-digital “Block V” receivers (also known as the Advanced Receivers [147]) installed in the early 1990s that had, under ideal circumstances, an S-band system noise of only 12.9 K, and a receiver loop bandwidth of 0.1 Hz.

Together with the enlarged 70 m antenna, these improved receivers made it possible to receive the signal of Pioneer 10 at 83 AU with a bit error rate of ∼ 1%. This error rate was reduced further by the convolutional code in use by Pioneer 10 and 11, which amounted to a 3.8 dB improvement in the signal-to-noise ratio, resulting in an error rate of about one in 104 bits.

During planetary encounters, the rapidly changing velocity of the spacecraft can result in a Doppler shift of its received frequency. During Pioneer 11’s close encounter with Jupiter, this rate could be as high as df/dt = 0.52 Hz/s [47]. Such a rapid change in received frequency can exceed the capabilities of the DSN closed loop receivers to remain “in lock”. To maintain continuous communication with such spacecraft, a “ramping” technique was implemented that allowed the tuned frequency of the DSN receiver to follow closely the predicted frequency of the spacecraft’s transmission [182]. This ramping technique was used successfully during the Pioneer 10 and 11 planetary encounters. Later it became routine operating procedure for the DSN. (As late as Pioneer 11’s 1979 encounter with Saturn, ramp frequencies at DSS-62 were tuned manually by relays of operators, as an automatic tuning capability was not yet available [241]).

3.1.4 DSN transmitters

To maintain continuous communication with distant spacecraft, the DSN must be able to transmit a signal to the spacecraft. The 70 m stations of the DSN are equipped with transmitters with a maximum transmitting power of 400 kW in the S-bandFootnote 12. This is sufficient power to maintain continuous communication with distant spacecraft even when the spacecraft are not oriented favorably relative to the Earth, and must use a low-gain omnidirectional antenna to receive ground commands.

Just like its receivers, the DSN’s transmitters are also capable of ramping. Ramped transmissions are necessary during planetary encounters in order to ensure that the spacecraft’s closed loop receiver remains in lock even as its line-of-sight velocity relative to the Earth is changing rapidlyFootnote 13.

3.1.5 Data communication

Results of Pioneer radio observations were packaged in data files (see Figure 3.5). Initially, these data files were transcribed to magnetic tape and delivered physically to the project site where they were processed. The present-day DSN uses electronic ground communication networks for this purpose.

3.2 Acquisition of radiometric Doppler data

Radiometric observations are performed by the DSN stations located at several DSN sites around the world [235, 240]. Routine radiometric tracking and navigation of the Pioneer 10 and 11 spacecraft was performed using Doppler observations. The Doppler extraction process is schematically depicted in Figure 3.3. A reference signal of a known frequency (usually a frequency close to that of the original transmission) is mixed with the received signal. The resulting beat frequency is than measured by a frequency counter. The count over a given time interval is recorded as the Doppler observation.

Figure 3.3
figure 24

The Doppler extraction process. Adapted from [374].

Receiving stations of the DSN are equipped with ultra-stable oscillators, allowing very precise measurements of the frequency of a received signal. One way to accomplish this measurement is by comparing the frequency of the received signal against a reference signal of known frequency, and count the number of cycles of the resulting beat frequency for a set period of time. This Doppler count is then stored in a file for later analysis.

The operating principles of the DSN evolved over time, and modern receivers may not utilize a beat frequency. However, all Pioneer data was stored using a format that incorporates an actual or simulated reference frequency.

Below we discuss the way Doppler observables are formed at the radio-science subsystem of the DSN. This description applies primarily to Block III and Block IV receivers, which were used for Pioneer radiometric observations throughout most of the Pioneer 10 and 11 missions.

3.2.1 DSN frequency and timing systemFootnote 14

Present day radiometric tracking of spacecraft requires highly accurate timing and frequency standards at tracking stations. For a two-way Doppler experiment, for instance, that involves transmitting a signal to a distant spacecraft and then receiving a response perhaps several hours later, long-term stability of tracking station oscillators is essential. Furthermore, if more than one tracking station is involved in an observation (e.g., “three-way” Doppler in which case one station is used to uplink a signal to the spacecraft, and another station receives the spacecraft’s response), frequency and clock synchronization between the participating tracking stations must be very precise [38].

Originally, the DSN used crystal oscillators. In the 1960s, these were replaced by rubidium or cesium oscillators that offered much improved frequency stability, and made precision radio-tracking of distant spacecraft possible. Rubidium and cesium oscillators offered a frequency stability of one part in ∼ 1012 and one part in ∼ 1013, respectively [241], over typical Doppler counting intervals of 102–103 seconds. (This is in agreement with the expected Allan deviation for the S-band signals.)

A further improvement occurred in the 1970s, when the rubidium and cesium oscillators were in turn replaced by hydrogen masers, which offered another order of magnitude improvement in frequency stability [241]. Today, the DSN’s frequency and timing system (FTS) is the source for the high accuracy mentioned above (see Figure 3.4). At its center is a hydrogen maser that produces a precise and stable reference frequency. These devices have Allan deviations (see Section 5.3.6) of approximately 3 × 10−15 to 1 × 10−15 for integration times of 102 to 103 seconds, respectively [38, 374]. These masers are good enough that the quality of Doppler-measurement data is limited by thermal or plasma noise, and not by the inherent instability of the frequency references.

Figure 3.4
figure 25

Block diagram of the DSN baseline configuration as used for radio Doppler tracking of the Pioneer 10 and 11 spacecraft. Adapted from [27, 101]. (IF stands for Intermediate Frequency.)

However, for two-way Doppler analysis, the relevant quantity is the stability of the station’s frequency standard during the round-trip light travel time. During the 1980s, hydrogen maser frequency standards were required to have Allan deviations of less than 10−14 over time scales of 3–12 hours [80, 123]. This corresponds to a Doppler frequency noise of 2 × 10−5 Hz.

For three-way Doppler analysis, where the transmitting and receiving stations are different, the frequency offset between stations is the relevant quantity. During the 1980s, station frequencies were controlled to be the same, to a fractional error of 10−12 [308, 356]. This corresponds to a Doppler frequency bias of up to 2 × 10−3 Hz between station pairs. Station keepers did maintain frequency offset knowledge to a tighter level, but for the most part this knowledge is not available to analysts today.

Three-way Doppler analysis is weakly sensitive to clock offsets between stations. By 1968, the operational technique for time synchronization was the “Moon Bounce Time Synch” technique, in which a precision-timed X-band signal from DSS-13 (which served as master timekeeper) was transmitted to overseas stations by way of a lunar reflection, achieving a clock accuracy of 5 µs between stations [241]. Such timing offsets would produce Doppler errors of less than 10−6 Hz. Later, DSN stations were synchronized utilizing the Global Positioning Satellite (GPS) network, achieving a synchronization accuracy of 1 µs or better [374].

3.2.2 The digitally controlled oscillator and exciter

Using the highly stable output from the FTS, the digitally controlled oscillator (DCO), through digitally controlled frequency multipliers, generates the Track Synthesizer Frequency (TSF) of ∼ 22 MHz. This is then sent to the Exciter Assembly. The Exciter Assembly multiplies the TSF by 96 to produce the S-band carrier signal at ∼ 2.2 GHz. The signal power is amplified by Traveling Wave Tubes (TWT) for transmission. If ranging data are required, the Exciter Assembly adds the ranging modulation to the carrier. The DSN tracking system has undergone many upgrades during the 34 years of tracking Pioneer 10. During this period internal frequencies have changed (see Section 3.1.1).

This S-band frequency is sent to the antenna where it is amplified and transmitted to the spacecraft. The onboard receiver tracks the up-link carrier using a phase lock loop. To ensure that the reception signal does not interfere with the transmission, the spacecraft (e.g., Pioneer) has a turnaround transponder with a ratio of 240/221 in the S-band. The spacecraft transmitter’s local oscillator is phase locked to the up-link carrier. It multiplies the received frequency by the above ratio and then re-transmits the signal to Earth.

3.2.3 Receiver and Doppler extractor

When the signal reaches the ground, the receiver locks on to the signal and tunes the Voltage Control Oscillator (VCO) to null out the phase error. The signal is sent to the Doppler Extractor. At the Doppler Extractor the current transmitter signal from the Exciter is multiplied by 240/221 (or 880/241 in the X-band) and a bias of 1 MHz for S-band or 5 MHz for X-band is added to the Doppler. The Doppler data is no longer modulated at S-band but has been reduced as a consequence of the bias to an intermediate frequency of 1 or 5 MHz.

The transmitter frequency of the DSN is a function of time, due to ramping and other scheduled frequency changes. When a two-way or three-way (see Section 3.2.5) Doppler measurement is performed, it is necessary to know the precise frequency at which the uplink signal was transmitted. This, in turn, requires knowledge of the exact light travel time to and from the spacecraft, which is available only when the position of the spacecraft is determined with precision. For this reason, DSN transmitter frequencies are recorded separately so that they can be accounted for in the orbit determination programs that we discuss in Section 5.

3.2.4 Metric data assembly

The intermediate frequency (IF) of 1 or 5 MHz with a Doppler modulation is sent to the Metric Data Assembly (MDA). The MDA consists of computers and Doppler counters where continuous count Doppler data are generated. From the FTS a 10 pulse per second signal is also sent to the MDA for timing. At the MDA, the IF and the resulting Doppler pulses are counted at a rate of 10 pulses per second. At each tenth of a second, the number of Doppler pulses is recorded. A second counter begins at the instant the first counter stops. The result is continuously-counted Doppler data. (The Doppler data is a biased Doppler of 1 MHz, the bias later being removed by the analyst to obtain the true Doppler counts.) The range data (if present) together with the Doppler data are sent separately to the Ranging Demodulation Assembly. The accompanying Doppler data is used to “rate-aid” (i.e., to “freeze” the range signal) for demodulation and cross correlation.

3.2.5 Radiometric Doppler data

Doppler data is the measure of the cumulative number of cycles of a spacecraft’s carrier frequency received during a user-specified count interval. The exact precision to which these measurements can be carried out is a function of the received signal strength and station electronics, but it is a small fraction of a cycle. Raw Doppler data is generated at the tracking station and delivered via a DSN interface to customers.

When the measured signal originates on the spacecraft, the resulting Doppler data is called one-way or F1 data. In order for such data to be useful for precision navigation, the spacecraft must be equipped with a precision oscillator on board. The Pioneer 10 and 11 spacecraft had no such oscillator. Therefore, even though a notable amount of F1 Doppler data was collected from these spacecraft, these data are not usable for precision orbit determination.

An alternative to one-way Doppler involves a signal generated by a transmitter on the Earth, which is received and then returned by the spacecraft. The Pioneer 10 and 11 spacecraft were equipped with a radio communication subsystem that had the capability to operate in “coherent” mode, a mode of operation in which the return signal from the spacecraft is phase-locked to a signal received by the spacecraft from the Earth. In this mode of operation, the precision of the frequency measurement is not limited by the stability of equipment on board the spacecraft, only by the frequency stability of ground-based DSN stations.

When the signal is transmitted from, and received by, the same station, the measurement is referred to as a two-way or F2 Doppler measurement; if the transmitting and receiving stations differ, the measurement is a three-way (F3) Doppler measurement.

Knowing the frequency of a signal received at a precise time at a precisely known location is only half the story in the case of two-way or three-way Doppler data: information must also be known about the time and location of transmission and the frequency of the transmitted signal.

The frequency of the transmitter, or the frequency of the receiver’s reference oscillator may not be fixed. In order to achieve better quality communication with spacecraft the velocity of which varies with respect to ground stations, a technique called ramping has been implemented at the DSN. When a frequency is ramped, it is varied linearly starting with a known initial frequency, at a known rate of frequency change over unit time.

Thus, a Doppler data point is completely characterized by the following:

  • The location of the receiving station,

  • The time of reception,

  • The reference frequency of the receiver,

  • Receiving station ramp information,

  • The length of the Doppler count interval,

  • The beat frequency (Doppler) count,

  • Transmission frequency,

and, for two- and three-way signals only,

  • The location of the transmitting station,

  • Transmitting station amp information.

Below we discuss the radiometric Doppler data formats that were used to support navigation of the Pioneer 10 and 11 spacecraft.

3.2.6 Pioneer Doppler data formats

The Pioneer radiometric data was received by the DSN in “closed-loop” mode, i.e., it was tracked with phase lock loop hardware. (“Open loop” data is recorded to tape but not tracked by phase lock loop hardware.) There are basically two types of data: Doppler (frequency) and range (time of flight), recorded at the tracking sites of the DSN as a function of UT Ground Received Time [103]. During their missions, the raw radiometric tracking data from Pioneer 10 and 11 were received originally in the form of Intermediate Data Record (IDR) tapes, which were then processed into special binary files called Archival Tracking Data Files (ATDF, format TRK-2-25 [95]), containing Doppler data from the standard DSN tracking receivers (Figure 3.5)Footnote 15. Note that the “closed-loop” data constitutes the ATDFs that were used in [27]. Figure 3.5 shows a typical tracking configuration for a Pioneer-class mission and corresponding data format flow.

Figure 3.5
figure 26

Typical tracking configuration for a Pioneer-class mission and corresponding data format flow [397].

ATDFs are files of radiometric data produced by the Network Operations Control Center (NOCC) Navigation Subsystem (NAV) (see Figure 3.4). They are derived from Intermediate Data Records by NAV and contain all radiometric measurements received from the DSN station including signal levels (AGC = automatic gain control in dB), antenna pointing angles, frequency (often referred to simply as “Doppler”), range, and residuals. Each ATDF consists of all tracking data types used to navigate a particular spacecraft (Pioneers had only Doppler data type) and typically include Doppler, range and angular types (in S-, X-, and L-frequency bands), differenced range versus integrated Doppler, programmed frequency data, pseudo-residuals, and validation data. (Unfortunately there was no range capability implemented on Pioneer 10 and 11 [24, 27]. Early in the mission, JPL successfully simulated range data using the “ramped-range” technique [182]. In this method, the transmitter frequency is ramped up and down with a known pattern. One round trip light time later, this same pattern appears at the DSN receiver, and can be used to solve for the round trip light travel time. Later in the mission, this technique became unusable because the carrier tracking loop bandwidth required to detect the carrier became too narrow to track the ramped frequency changes.) Also, ATDFs contain data for a single spacecraft, for one or more ground receiving stations, and for one or more tracking passes or days.

After a standard processing at the Radio Metric Data Conditioning group (RMDC) of JPL’s Navigation and Mission Design Section, ATDFs are transformed into Orbit Data Files (ODF, format TRK-2-18 [96]). A program called STRIPPER is used to produce the ODFs that are, at this point, the main product that is distributed to the end users for their orbit determination needs (see more discussion on the conversion process in [27]). At JPL, after yet additional processing, these ODFs are used to produce sequentially formatted input/output files in NAVIO format that is used by navigators while working with the JPL Orbit Determination Program. (Note that the NAVIO input/output file format is used only at JPL; other orbit determination programs convert ODFs to their particular formats.)

Each ODF physical record is 2016 32-bit words in length and consists of 224 9-word logical records per data block. The ODF records are arranged in a sequence that consists of one file label record, one file identifier logical record, orbit data logical records in time order, ramp data logical records in time order, clock offset data logical records in time order, data summary logical records in time order and software/hardware end-of-file markers. Bit lengths of data fields are variable and cross word boundaries. An ODF usually contains most types of records, but may not have them all. The first record in each of the 7 primary groups is a header record; depending on the group, there may be from zero to many data records following each header. For further details, see Appendix B.

3.3 Available Doppler data

The Pioneer 10 and 11 spacecraft were in space for more than three decades. The original 2002 study [27] of the Pioneer anomaly (see Section 5) used approximately 11.5 years of Pioneer 10 and 3.5 years of Pioneer 11 Doppler data. These data points were obtained from the late phases of the respective missions, when the spacecraft were far from the Earth and the Sun.

An effort was launched at JPL in June 2005 to recover more Pioneer Doppler data, preferably the entire Doppler mission records, if possible: almost 30 years of Pioneer 10 and 20 years of Pioneer 11 Doppler data, most of which was never used previously in the investigation of the anomalous acceleration. This task proved much harder than anticipated, due to antiquated file formats, missing data, and corrupted files (see discussion below and [397]).

Below we discuss these two sets of Pioneer 10 and 11 Doppler data.

3.3.1 Pioneer Doppler data available in 2002

The anomalous acceleration of the Pioneer spacecraft was first reported in 1998 [24]. This effort utilized Pioneer 10 data from 1987 to 1995, and a shorter span of Pioneer 11 data to obtain accelerations of aP10 = (8.09±0.20) × 10−10 m/s2, and aP11 = (8.56±0.11) × 10−10 m/s2, respectively, using JPL’s Orbit Determination Program (ODP). The Pioneer 10 result was also verified by the Aerospace Corporation’s Compact High Accuracy Satellite Motion Program (CHASMP), which yielded aP10 = (8.65 ± 0.03) × 10−10 m/s2. The errors quoted are the statistical formal errors produced by the fitting procedure.

In 2002, JPL published the results of a study that, to this date, remains the definitive result on the Pioneer anomaly [27] (see Section 5). In this study, a significantly expanded set of Doppler data was utilized. For Pioneer 10, the data set covered the period between January 3, 1987 and July 22, 1998. This corresponds to heliocentric distances between 40 and 70.5 AU. The data set contained 19,403 Doppler data points. For Pioneer 11, the data set was smaller: the trajectory of the spacecraft between January 5, 1987 and October 1, 1990 was covered, corresponding to heliocentric distances between 22.42 and 31.7 AU, with 10,252 Doppler data points.

The 2002 JPL study was also the first to combine Pioneer 10 and Pioneer 11 results. The study attempted to estimate realistic errors, e.g., errors due to physical or computational systematic effects. This approach resulted in the value of aP = (8.74±1.33) × 10−10 m/s2, which is now the widely quoted “canonical” value of the anomalous acceleration of Pioneer 10 and 11.

3.3.2 The extended Pioneer Doppler data set

The recovery of radiometric Doppler data for a mission with such a long duration was never attempted before. It presents unique challenges, as a result of changing data formats, changes in navigational software and supporting hardware, changes in the configuration of the DSN (new stations built, old ones demolished, relocated, or upgraded), and the loss of people [397]. Even physically locating the data proved to be a difficult task, as incomplete holdings were scattered among various archives. Nonetheless, as of November 2009, the transfer of the available Pioneer Doppler data to modern media formats has been completed.

Initially, the following sources for Doppler data were considered:

  • Archived tracking files at JPL and the Deep Space Network;

  • Files archived by JPL researchers at the NSSDC (i.e., National Space Science Data Center);

  • Data segments available from individual researchers at JPL.

During these data collection efforts, multiple serious problems were encountered (see details in [397]), including

  • The files were in several different formats, including old (e.g., “Type 66” [401]) formats that are no longer used, nor very well documented, and for which no data conversion tools exist;

  • The files were often missing critical information; for instance, tracking data may have been recorded for the spacecraft, but critically important ramp (see Section 3.1.3 and also [182]) data for the corresponding transmitting stations may have been lost;

  • The files were not processed with a consistent strategy; for instance, some files contained Doppler frequency data that was corrected for the spacecraft’s spin, whereas other files did not include such corrections;

  • Some files were corrupted but recoverable; for instance, writing fixed size records using inappropriate software tools may have introduced spurious bytes into the data in a manner that can be corrected;

  • Some files were corrupted in an unrecognizable manner and had to be discarded.

Despite the unanticipated complexities, as of late 2009 the transfer of the available Pioneer Doppler data to modern media formats has been completed. Initial analysis of these data is under way, and it appears that while Pioneer 10 data prior to February 1980 is not usable, coverage is nearly continuous from that data until the end of mission. Similarly, for Pioneer 11, good data is available from mid-1978 until the loss of coherent mode in late 1990. Further details will be reported as appropriate upon the conclusion of this initial data analysis.

To summarize, there exists more than 20 years of Pioneer 10 and more than 10 years of Pioneer 11 data, a significant fraction of which had never been well studied for the purposes of anomaly investigation. This new, expanded data set may make it possible to answer questions concerning the constancy and direction of the anomalous acceleration.

3.4 Doppler observables and data preparation

The Doppler observable can be predicted if the spacecraft’s orbit is known. Given known initial conditions and the ephemerides of gravitating sources, a dynamic model can be constructed that yields predictions of the spacecraft’s position as a function of time. An observational model can account for the propagation of the signal, allowing a computation of the received signal frequency at a ground station of known terrestrial location. The difference between the calculated and observed values of the received frequency is known as the Doppler residual. If this residual exceeds acceptable limits, the dynamic model or observational model must be adjusted to account for the discrepancy. Once the model is found to be sufficiently accurate, it can also be used to plan the spacecraft’s future trajectory (see Figure 4.1 and relevant discussion in Section 4).

Figure 4.1
figure 27

Schematic overview of the radio navigation process. Adapted from [374].

Various radio tracking strategies are available for determining the trajectory parameters of interplanetary spacecraft. However, radio tracking Doppler and range techniques are the most commonly used methods for navigational purposes. (Note that Pioneers did not have a range observable; all the navigational data is in the form of Doppler observations.) The position and velocities of the DSN tracking stations must be known to high accuracy. The transformation from a Earth fixed coordinate system to the International Earth Rotation Service (IERS) Celestial System is a complex series of rotations that includes precession, nutation, variations in the Earth’s rotation (UT1-UTC) and polar motion.

Calculations of the motion of a spacecraft are made on the basis of the range time-delay and/or the Doppler shift in the signals. This type of data was used to determine the positions, the velocities, and the magnitudes of the orientation maneuvers for the Pioneer spacecraft.

Theoretical modeling of the group delays and phase delay rates are done with the orbit determination software we describe in Section 4.

3.4.1 Doppler experimental techniques and strategy

In Doppler experiments a radio signal transmitted from the Earth to the spacecraft is coherently transponded and sent back to the Earth. Its frequency change is measured with great precision, using the hydrogen masers at the DSN stations. The observable is the DSN frequency shiftFootnote 16

$$\Delta \nu (t) = {\nu _0}{1 \over c}{{d\ell} \over {dt}},$$

where is the overall optical distance (including diffraction effects) traversed by a photon in both directions. (In the Pioneer Doppler experiments, the stability of the fractional drift at the S-band is on the order of Δν/ν0 ≃ 10−12, for integration times on the order of 103 s.) Doppler measurements provide the “range rate” of the spacecraft and therefore are affected by all the dynamical phenomena in the volume between the Earth and the spacecraft.

Expanding upon what was discussed in Section 3.2, the received signal and the transmitter frequency (both are at S-band) as well as a 10 pulse per second timing reference from the FTS are fed to the Metric Data Assembly (MDA). There the Doppler phase (difference between transmitted and received phases plus an added bias) is counted. That is, digital counters at the MDA record the zero crossings of the difference (i.e., Doppler, or alternatively the beat frequency of the received frequency and the exciter frequency). After counting, the bias is removed so that the true phase is produced.

The system produces “continuous count Doppler” and it uses two counters. Every tenth of a second, a Doppler phase count is recorded from one of the counters. The other counter continues the counts. The recording alternates between the two counters to maintain a continuous unbroken count. The Doppler counts are at 1 MHz for S-band or 5 MHz for X-band. The wavelength of each S-band cycle is about 13 cm. Dividers or “time resolvers” further subdivide the cycle into 256 parts, so that fractional cycles are measured with a resolution of 0.5 mm. This accuracy can only be maintained if the Doppler is continuously counted (no breaks in the count) and coherent frequency standards are kept throughout the pass. It should be noted that no error is accumulated in the phase count as long as lock is not lost. The only errors are the stability of the hydrogen maser and the resolution of the “resolver.”

Consequently, the JPL Doppler records are not frequency measurements. Rather, they are digitally counted measurements of the Doppler phase difference between the transmitted and received S-band frequencies, divided by the count time.

Therefore, the Doppler observables to which we will refer have units of cycles per second or Hz. Since total count phase observables are Doppler observables multiplied by the count interval Tc, they have units of cycles. The Doppler integration time refers to the total counting of the elapsed periods of the wave with the reference frequency of the hydrogen maser. The usual Doppler integrating times for the Pioneer Doppler signals refers to the data sampled over intervals of 10 s, 60 s, 600 s, 660 s, or 1980 s.

In order to acquire Doppler data, the user must provide a reference trajectory and information concerning the spacecraft’s RF system to JPL’s Deep Space Mission System (DSMS), to allow for the generation of pointing and frequency predictions. The user specified count interval can vary from 0.1 s to tens of minutes. Absent any systematic errors, the precision improves as the square root of the count interval. Count times of 10 to 60 seconds are typical [97], as well as intervals of ∼ 2000 s, which is an averaging interval located at the minimum of the Allan variance curve for hydrogen masers. The average rate of change of the cycle count over the count interval expresses a measurement of the average velocity of the spacecraft in the line between the antenna and the spacecraft. The accuracy of Doppler data is quoted in terms of how accurate this velocity measurement is over a 60 second count.

It is also possible to infer the position in the sky of a spacecraft from the Doppler data. This is accomplished by examining the diurnal variation imparted to the Doppler shift by the Earth’s rotation. As the ground station rotates underneath a spacecraft, the Doppler shift is modulated by a sinusoid. The sinusoid’s amplitude depends on the declination angle of the spacecraft and its phase depends upon the right ascension. These angles can therefore be estimated from a record of the Doppler shift that is (at least) of several days duration. This allows for a determination of the distance to the spacecraft through the dynamics of spacecraft motion using standard orbit theory contained in the orbit determination programs.

3.4.2 Data preparation

In an ideal system, all scheduled observations would be used in determining parameters of physical interest. However, there are inevitable problems that occur in data collection and processing that corrupt the data. So, at various stages of the signal processing one must remove or “edit” corrupted data. Thus, the need arises for objective editing criteria. Procedures have been developed, which attempt to excise corrupted data on the basis of objective criteria. There is always a temptation to eliminate data that is not well explained by existing models, to thereby “improve” the agreement between theory and experiment. Such an approach may, of course, eliminate the very data that would indicate deficiencies in the a priori model. This would preclude the discovery of improved models.

In the processing stage that fits the Doppler samples, checks are made to ensure that there are no integer cycle slips in the data stream that would corrupt the phase. This is done by considering the difference of the phase observations taken at a high rate (10 times a second) to produce Doppler. Cycle slips often are dependent on tracking loop bandwidths, the signal-to-noise ratios, and predictions of frequencies. Blunders due to out-of-lock can be determined by looking at the original tracking data. In particular, cycle slips due to loss-of-lock stand out as a 1 Hz blunder point for each cycle slipped.

If a blunder point is observed, the count is stopped and a Doppler point is generated by summing the preceding points. Otherwise the count is continued until a specified maximum duration is reached. Cases where this procedure detected the need for cycle corrections were flagged in the database and often individually examined by an analyst. Sometimes the data was corrected, but nominally the blunder point was just eliminated. This ensures that the data is consistent over a pass. However, it does not guarantee that the pass is good, because other errors can affect the whole pass and remain undetected until the orbit determination is done.

To produce an input data file for an orbit determination program, JPL has a software package known as the Radio Metric Data Selection, Translation, Revision, Intercalation, Processing and Performance Evaluation Reporting (RMD-STRIPPER) program. As we discussed in Section 3.4.1, this input file has data that can be integrated over intervals with different durations: 10 s, 60 s, 600 s, 660 s, and 1980 s. This input orbit determination file obtained from the RMDC group is the data set that can be used for analysis. Therefore, the initial data file already contained some common data editing that the RMDC group had implemented through program flags, etc. The data set we started with had already been compressed to 60 s. So, perhaps there were some blunders that had already been removed using the initial STRIPPER program.

The orbit analyst manually edits the remaining corrupted data points. Editing is done either by plotting the data residuals and deleting them from the fit or plotting weighted data residuals. That is, the residuals are divided by the standard deviation assigned to each data point and plotted. This gives the analyst a realistic view of the data noise during those times when the data was obtained while looking through the solar plasma. Applying an “Nσ” (σ is the standard deviation) test, where N is the choice of the analyst (usually 4–10) the analyst can delete those points that lie outside the Nσ rejection criterion without being biased in his selection.

A careful analysis edits only very corrupted data; e.g., a blunder due to a phase lock loss, data with bad spin calibration, etc. If needed or desired, the orbit analyst can choose to perform an additional data compression of the original navigation data.

3.4.3 Data weighting

The Pioneers used S-band (∼ 2.2 GHz) radio signals to communicate with the DSN. The S-band data is available from 26 m, 70 m, and some 34 m antennas of the DSN complex (see baseline DSN configuration in the Figure 3.4). The dominant systematic error that can affect S-band tracking data is ionospheric transmission delays. When the spacecraft is located angularly close to the Sun, with Sun-Earth-spacecraft angles of less than 10 degrees, degradation of the data accuracy will occur. S-band data is generally unusable for Sun-Earth-spacecraft angles of less than 5 degrees.

Therefore, considerable effort has gone into accurately estimating measurement errors in the observations. These errors provide the data weights necessary to accurately estimate the parameter adjustments and their associated uncertainties. To the extent that measurement errors are accurately modeled, the parameters extracted from the data will be unbiased and will have accurate sigmas assigned to them. Typically, for S-band Doppler data one assigns a standard 1−σ uncertainty of 1 mm/s over a 60 s count time after calibration for transmission media effects.

A change in the DSN antenna elevation angle also directly affects the Doppler observables due to tropospheric refraction. Therefore, to correct for the influence of the Earth’s troposphere the data can also be deweighted for low elevation angles. The phenomenological range correction used in JPL’s analysis technique is given as

$$\sigma = {\sigma _{{\rm{nomianal}}}}\left({1 + {{18} \over {{{(1 + {\theta _E})}^2}}}} \right),$$

where σnominal is the basic standard deviation (in Hz) and θE is the elevation angle in degrees. Each leg is computed separately and summed. For Doppler the same procedure is used. First, Equation (3.8) is multiplied by \(\sqrt {60\;{\rm{s}}/{T_c}}\), where Tc is the count time. Then a numerical time differentiation of Equation (3.8) is performed. That is, Equation (3.8) is differenced and divided by the count time, Tc. (For more details on this standard technique see [237, 240].)

There is also the problem of data weighting for data influenced by the solar corona. This is discussed in Section 4.5.1.

3.4.4 Spin calibration of the data

The radio signals used by DSN to communicate with spacecraft are circularly polarized. When these signals are reflected from spinning spacecraft antennas a Doppler bias is introduced that is a function of the spacecraft spin rate. Each revolution of the spacecraft adds one cycle of phase to the up-link and the down-link. The up-link cycle is multiplied by the turn around ratio 240/221 so that the bias equals (1+240/221) cycles per revolution of the spacecraft.

For the Pioneer 10 and 11 spacecraft, high-accuracy spin data is available from the spacecraft telemetry. Due to the star sensor failure on board Pioneer 10 (see Section 2), once the spacecraft was more than ∼ 30 AU from the Sun, no on-board roll reference was available. Until mid-1993, a science instrument (the Infrared Photo-Polarimeter) was used as a surrogate star sensor, which allowed the accurate determination of the spacecraft spin rate; however, due to the lack of available electrical power on board, this instrument could not be used after 1993. However, analysts still could get a rough spin determination approximately every six months using information obtained from the conscan maneuvers. No spin determinations were made after 1995. However, the archived conscan data could still yield spin data at every maneuver time if such work was approved. Further, as the phase center of the main antenna is slightly offset from the spin axis, a very small (but detectable) sine-wave signal appears in the high-rate Doppler data. In principle, this could be used to determine the spin rate for passes taken after 1993, but it has not been attempted.

The changing spin rates of Pioneer 10 and 11 can be an indication of gas leaks, which can also be a source of unmodeled accelerations. We discussed this topic in more detail in Section 2.3.7.

3.5 Pioneer telemetry data

Telemetry received from the Pioneer 10 and 11 spacecraft was supplied to the Pioneer project by the DSN in the form of Master Data Records (MDRs). MDRs contained all information sent by the spacecraft to the ground, and some information about the DSN station that received the data. The information in the MDRs that was sent by the spacecraft included engineering telemetry as well as scientific observations.

The Pioneer project used engineering data extracted from the MDRs to monitor and control the spacecraft, while scientific data, also extracted from the MDRs, was converted into formats specific to each experiment and supplied to the experimenter groups.

Far beyond the original expectationsFootnote 17, telemetry is now seen to be of value for the investigation of the Pioneer anomaly, as the MDRs, specifically the telemetry data contained therein, are helpful in the construction of an accurate model of the spacecraft during their decades long journey, including a precise thermal profile, the time history of propulsion system activation and usage, and many other potential sources of on-board disturbances. After recent recovery efforts [397], this data is available for investigation.

3.5.1 MDR data integrity and completeness

The total amount of data stored in these MDR files is approximately 40 GB [397]. According to the original log sheets that record the transcription from tape to magneto-optical media, only a few days worth of data is missing, some due to magnetic tape damage. One notable exception is the Jupiter encounter period of Pioneer 10. According to the transcription log sheets, DOY 332–341 from 1973 were not available at the time when the magnetic tapes on which the MDRs were originally stored were transcribed to more durable magneto-optical media.

Other significant periods of missing data are listed in Table 3.5. It is not known why these records are not present, except that we know that very few days are missing due to unreadable media (i.e., the cause is missing, not damaged, tapes.)

Table 3.5 Pioneer 10 and 11 missing MDRs (periods of missing data shorter than 1–2 days not shown.)

So, the record is fairly complete. But how good is the data? Over forty billion bytes were received by the DSN, processed, copied to tape, copied from tape to magneto-optical disks, then again copied over a network connection to a personal computer. It is not inconceivable that the occasional byte was corrupted by a transmission or storage error. There are records that contain what is apparently bogus data, especially from the later years of operation. This, plus the fact that the record structure (e.g., headers, synchronization sequences) is intact suggests reception errors as the spacecraft’s signal got weaker due to increasing distance, and not copying and/or storage errors.

The MDRs contain no error detection or error correction code, so it is not possible to estimate the error rate. However, it is likely to be reasonably low, since the equipment used for storage and copying is generally considered very reliable. Furthermore, any errors would likely show up as random noise, and not as a systemic bias. In this regard, the data should generally be viewed to be of good quality insofar as the goal of constructing an engineering profile of the spacecraft is considered.

3.5.2 Interpreting the data

MDRs are a useless collection of bits unless information is available about their structure and content. Fortunately, this is the case in the case of the Pioneer 10 and Pioneer 11 MDRs.

The structure of an MDR is shown in Appendix C (see also [402]). The frame at the beginning and at the end of each 1344-bit record contained information about the DSN station that received the data, and included a timestamp, data quality and error indicators, and the strength of the received signal. The middle of the record was occupied by as many as four consecutive data frames received by the spacecraft.

The MDR header is followed by four data frames (not all four frames may be used, but they are all present) of 192 bits each. Lastly, an additional 8 words of DSN information completes the record. The total length of an MDR is thus 42 words of 32 bits each.

The 192-bit data frames are usually interpreted as 64 3-bit words or, alternatively, as 32 6-bit words. The Pioneer project used many different data frame formats during the course of the mission. Some formats were dedicated to engineering telemetry (accelerated formats). Other formats are science data formats, but still contain engineering telemetry in the form of a subcommutator: a different engineering telemetry value is transmitted in each frame, and eventually, all telemetry values are cycled through.

The Pioneer spacecraft had a total of 128 6-bit words reserved for engineering telemetry. Almost all these values are, in fact, used. A complete specification of the engineering telemetry values can be found in Section 3.5 (“Data Handling Subsystem”) of [292]. When engineering telemetry was accelerated to the main frame rate, four different record formats (C-1 through C-4) were used to transmit telemetry information. When the science data formats were in use, an area of the record was reserved for a subcommutator identifier and value.

The formats are further complicated by the fact that some engineering telemetry values appear only in subcommutators, whereas others only appear at the accelerated (main frame) rate.

In the various documentation packages, engineering data words are identified either by mnemonic, by the letter ‘C’ followed by a three-digit number that runs from 1 through 128, or most commonly, by the letter ‘C’ followed by a digit indicating which ‘C’ record (C-1 through C-4) the value appears in, and a two-digit number between 1 and 32: for instance, C-201 means the first engineering word in the C-2 record.

3.5.3 Available telemetry information

Pioneer 10 and 11 telemetry data is very relevant for a study of on-board systematics. The initial studies of the Pioneer anomaly [24, 27, 194, 390] and several subsequent papers [28, 391, 392, 393] had emphasized the need for a very detailed investigation of the on-board systematics. Other researchers also focused their work on the study of several on-board generated mechanisms that could contribute to an anomalous acceleration of the spacecraft [164, 245, 327]. Most of these investigations of on-board systematics were not very precise. This was due to a set of several reasons, one of them is insufficient amount of actual telemetry data from the vehicles. In 2005, this picture changed dramatically when this critical information became available.

Table 3.6. summarizes all available telemetry values in the C (engineering) and E (science) telemetry formats. The MDRs also contain a complete set of science readings that were telemetered in the A, B, and D formats.

Table 3.6 Available parameter set that may be useful for the Pioneer anomaly investigation.

As this table demonstrates, telemetry readings can be broadly categorized as temperature, voltage, current readings; other analog readings; various binary counters, values, and bit fields; and readings from science instruments. Temperature and electrical readings are of the greatest use, as they help to establish a detailed thermal profile of the spacecrafts’ major components. Some binary readings are useful; for instance, thruster pulse count readings help to understand maneuvers and their impact on the spacecrafts’ trajectories. It is important to note, however, that some readings may not be available and others may not be trusted. For example, thruster pulse count readings are only telemetered when the spacecraft is commanded to send readings in accelerated engineering formats; since these formats were rarely used late in the mission, we may not have pulse count readings for many maneuvers. Regarding reliability, we know from mission status reports about the failure of the sun and star sensors; these failures invalidate many readings from that point onward. Thus it is important to view telemetry readings in context before utilizing them as source data for our investigation.

On-board telemetry not only gives a detailed picture of the spacecraft and its subsystems, but this picture is redundant: electrical, thermal, logic state and other readings provide means to examine the same event from a multitude of perspectives.

In addition to telemetry, there exists an entire archive of the Pioneer Project documents for the period from 1966 to 2003. This archive contains all Pioneer 10 and 11 project documents discussing the spacecraft and mission design, fabrication of various components, results of various tests performed during fabrication, assembly, pre-launch, as well as calibrations performed on the vehicles; and also administrative documents including quarterly reports, memoranda, etc. Information on most of the maneuver records, spin rate data, significant events of the craft, etc. is also available.

4 Navigation of the Pioneer Spacecraft

Modern radio tracking techniques made it possible to explore the gravitational environment in the solar system up to a level of accuracy never before possible. The two principal forms of celestial mechanics experiments that were used involve planets (e.g., passive radar ranging) and Doppler and range measurements with interplanetary spacecraft [19, 29]. This work was motivated by the desire to improve the ephemerides of solar system bodies and knowledge of solar system dynamics.

The main objective of spacecraft navigation is to determine the present position and velocity of a spacecraft and to predict its future trajectory. This is usually done by measuring changes in the spacecraft’s radio signal and then, using those measurements, correcting (fitting and adjusting) the predicted spacecraft trajectory.

In this section we discuss the theoretical foundations that are used for the analysis of tracking data from interplanetary spacecraft. We describe the basic physical models used to determine a trajectory.

4.1 Models for gravitational forces acting on a spacecraft

The primary force acting on a spacecraft in deep space is the force of gravity, specifically, the gravitational attraction of the Sun and, to a lesser extent, planetary bodies. When the spacecraft is in deep space, far from the Sun and not in the vicinity of any planet, these sources of gravity can be treated as Newtonian point sources. However, when the spacecraft is in the vicinity of a massive body, corrections due to general relativity as well as the finite extent and mass distribution of the body in question must be considered.

Of particular interest is the possibility that a celestial mechanics experiment might help distinguish between different theories of gravity. There exists a method, the parametrized post-Newtonian (PPN) formalism, that allows one to describe various metric theories of gravity up to \({\mathcal O({c^{- 5}})}\) using a common parameterized metric. As this formalism forms the basis of modern spacecraft navigational codes, in the following, we provide a brief summary.

4.1.1 The parametrized post-Newtonian formalism

In 1922, Eddington [113] developed the first parameterized generalization of Einstein’s theory, expressing components of the metric tensor in the form of a power series of the Newtonian potential. This phenomenological parameterization has since been developed into what is known as the parametrized post-Newtonian (PPN) formalism [230, 264, 265, 266, 267, 373, 412, 413, 414, 415, 416, 417, 419]. This formalism represents the metric tensor for slowly moving bodies and weak gravitational fields. The formalism is valid for a broad class of metric theories, including general relativity as a unique case. The parameters that appear in the PPN formalism are individually associated with various symmetries and invariance properties of the underlying theory (see [417] for details).

The full PPN formalism has 10 parameters [418]. However, if one assumes that Lorentz invariance, local position invariance and total momentum conservation hold, the metric tensor for a system of N point-like gravitational sources in four dimensions at the spacetime position of body i may be written as (see [389])

$$\begin{array}{*{20}c} {{g_{00}} = 1 - {2 \over {{c^2}}}\sum\limits_{j \neq i} {{{{\mu _j}} \over {{r_{ij}}}} + {{2\beta} \over {{c^4}}}{{\left[ {\sum\limits_{j \neq i} {{{{\mu _j}} \over {{r_{ij}}}}}} \right]}^2} - {{1 + 2\gamma} \over {{c^4}}}\sum\limits_{j \neq i} {{{{\mu _j}\dot r_j^2} \over {{r_{ij}}}}}} \quad \quad \quad} \\ {+ {{2(2\beta - 1)} \over {{c^4}}}\sum\limits_{j \neq i} {{{{\mu _j}} \over {{r_{ij}}}}} \sum\limits_{k \neq j} {{{{\mu _k}} \over {{r_{jk}}}} - {1 \over {{c^4}}}\sum\limits_{j \neq i} {{\mu _j}{{{\partial ^2}{r_{ij}}} \over {\partial {t^2}}} + {\mathcal O}({c^{- 5}}),}}} \\ {{g_{0\alpha}} = {{2(\gamma + 1)} \over {{c^3}}}\sum\limits_{j \neq i} {{{{\mu _j}{\bf{\dot r}}_j^\alpha} \over {{r_{ij}}}} + {\mathcal O}({c^{- 5}}),\quad (\alpha = 1 \ldots 3)\quad \quad \quad \quad}} \\ {{g_{\alpha \beta}} = - {\delta _{\alpha \beta}}\left({1 + {{2\gamma} \over {{c^2}}}\sum\limits_{j \neq i} {{{{\mu _j}} \over {{r_{ij}}}}}} \right) + {\mathcal O}({c^{- 5}}),\quad (\alpha, \,\beta = 1 \ldots 3)\quad} \\ \end{array}$$

where the indices 1 ≤ i, jN refer to the N bodies, rij is the distance between bodies i and j (calculated as ∣rjri∣ where ri is the spatial position vector of body i), μi is the gravitational constant for body i given as μi = Gmi, where G is the Newtonian gravitational constant and mi is the body’s rest mass. The Eddington-parameters β and γ have, in this special case, clear physical meaning: β represents a measure of the nonlinearity of the law of superposition of the gravitational fields, while γ represents the measure of the curvature of the spacetime created by a unit rest mass.

The Newtonian scalar gravitational potential in Equation (4.1) is given by the 1/c2 term in g00. Corrections of order 1/c4, parameterized by β and γ, are post-Newtonian terms. In the case of general relativity, β = γ = 1. One of the simplest generalizations of general relativity is the theory of Brans and Dicke [60] that contains, in addition to the metric tensor, a scalar field and an undetermined dimensionless coupling constant ω. Brans-Dicke theory yields the values β = 1, γ = (1 + ω)/(2 + ω) for the Eddington parameters. The value of β may be different for other scalar-tensor theories [88, 87, 389].

The PPN formalism is widely used in studies of relativistic gravitation [63, 64, 238, 239, 359, 387, 417]. The relativistic equations of motion for an N-body system are derived from the PPN metric using a Lagrangian formalism [387, 417], discussed below.

4.1.2 Relativistic equations of motion

Navigation of spacecraft in deep space requires computing a spacecraft’s trajectory and the compilation of spacecraft ephemeris: a file containing the position and velocity of the spacecraft as functions of time [389]. Spacecraft ephemerides are computed by orbit determination codes that utilize a numerical integrator in conjunction with various input parameters. These include an estimate of the spacecraft’s initial state vector (comprising its position and velocity vector), adopted constants (c, G, planetary mass ratios, etc.) and parameters that are estimated from fits to observational data (e.g., corrections to the ephemerides of solar system bodies).

The principal equations of motion used by orbit determination codes describe the relativistic gravitational acceleration in the presence of the gravitational field of multiple point sources that include the Sun, planets, major moons and larger asteroids [360]. These equations are derived from the metric Equation (4.1) using a Lagrangian formalism. The point source model is adequate in deep space when a spacecraft is traveling far from those sources. When the spacecraft is in the vicinity of a planet, ephemeris programs also compute corrections due to deviations from spherical symmetry in the planetary body, as well as the gravitational influences from the planet’s moons, if any.

The acceleration of body i due to the gravitational field of point sources, including Newtonian and relativistic perturbative accelerations [117, 387, 417], can be derived in the solar system barycentric frame in the form [27, 120, 237, 238, 240, 252]:

$$\begin{array}{*{20}c} {{{{\bf{\ddot r}}}_i} = \sum\limits_{j \neq i} {{{{\mu _j}({{\bf{r}}_j} - {{\bf{r}}_i})} \over {r_{ij}^3}}} \left\{{1 - {{2(\beta + \gamma)} \over {{c^2}}}\sum\limits_{l \neq i} {{{{\mu _l}} \over {{r_{il}}}} - {{2\beta - 1} \over {{c^2}}}} \sum\limits_{k \neq j} {{{{\mu _k}} \over {{r_{jk}}}} + \gamma {{\left({{{{{\dot r}_i}} \over c}} \right)}^2} + (1 + \gamma){{\left({{{{{\dot r}_j}} \over c}} \right)}^2}}} \right.} \\ {\left. {- {{2(1 + \gamma)} \over {{c^2}}}{{{\bf{\dot r}}}_i}{{{\bf{\dot r}}}_j} - {3 \over {2{c^2}}}{{\left[ {{{({{\bf{r}}_i} - {{\bf{r}}_j}){{{\bf{\dot r}}}_j}} \over {{r_{ij}}}}} \right]}^2} + {1 \over {2{c^2}}}({{\bf{r}}_j} - {{\bf{r}}_i}){{{\bf{\ddot r}}}_j}} \right\} + {{3 + 4\gamma} \over {2{c^2}}}\sum\limits_{j \neq i} {{{{\mu _j}{{{\bf{\ddot r}}}_j}} \over {{r_{ij}}}}}} \\ {+ {1 \over {{c^2}}}\sum\limits_{j \neq i} {{{{\mu _j}} \over {r_{ij}^3}}\left\{{[{{\bf{r}}_i} - {{\bf{r}}_j}] \cdot [(2 + 2\gamma){{{\bf{\dot r}}}_i} - (1 + 2\gamma){{{\bf{\dot r}}}_j}]} \right\}({{{\bf{\dot r}}}_i} - {{{\bf{\dot r}}}_j}) + {\mathcal O}({c^{- 4}}),}} \\ \end{array}$$

where the indices 1 ≤ j, k, lN refer to the N bodies and where k includes body i, whose motion is being investigated.

These equations can be integrated numerically to very high precision using standard techniques in numerical codes that are used to construct solar system ephemerides, for spacecraft orbit determination [240, 359, 387], and for the analysis of gravitational experiments in the solar system [388, 398, 417, 418, 420].

In the vicinity of a celestial body, one must also take into account that a celestial body is not spherically symmetric. Its gravitational potential can be modeled in terms of spherical harmonics. As Pioneers 10 and 11 both flew by Jupiter and Pioneer 11 visited Saturn, of specific interest to the navigation of theses spacecraft is the gravitational potential due to the oblateness of a planet, notably a gas giant. The gravitational potential due to the oblateness of planetary body i can be expressed using zonal harmonics in the form [240]:

$${U_{i{\rm{obl}}}} = - {{{\mu _i}} \over {\vert {{\bf{r}}_i} - {\bf{r}}\vert}}\sum\limits_{k = 1}^\infty {{{J_k^{(i)}a_i^k{P_k}(\sin \,\theta)} \over {\vert {{\bf{r}}_i} - {\bf{r}}{\vert ^k}}}},$$

where Pk(x) is the k-th Legendre polynomial in x, ai is the equatorial radius of planet i, θ is the latitude of the spacecraft relative to the planet’s equator, and \(J_k^{(i)}\) is the k-th spherical harmonic coefficient of planet i.

In order to put Equation (4.3) to use, first it must be translated into an expression for force by calculating its gradient Second, it is also necessary to express the position of the spacecraft in a coordinate system that is fixed to the planet’s center and equator.

4.2 Light times and time scales

The complex gravitational environment of the solar system manifests itself not just through its effects on the trajectory of a spacecraft or celestial body: the propagation of electromagnetic signals to or from an observing station on the Earth must also be considered. Additionally, proper timekeeping becomes an issue of significance: clocks that are in relative motion do not tick at the same rate, and changing gravitational potentials may also affect them.

4.2.1 Light time solution

The time it takes for a signal to travel between two locations in space in the gravitational environment of a massive point source with gravitational constant μ can be derived from the PPN metric Equation (4.1) in the form [240]:

$${t_2} - {t_1} = {{{r_{12}}} \over c} + (1 + \gamma){\mu \over {{c^3}}}\ln {{{r_1} + {r_2} + {r_{12}} + (1 + \gamma)\mu/{c^2}} \over {{r_1} + {r_2} - {r_{12}} + (1 + \gamma)\mu/{c^2}}} + {\mathcal O}({c^{- 5}}),$$

where t1 refers to the signal transmission time, and t2 refers to the reception time, r1,2 represent the distance of the point of transmission and point of reception, respectively, from the massive body, and r12 is the spatial separation of the points of transmission and reception. The terms proportional to μ/c2 are important only for the Sun and are negligible for all other bodies in the solar system.

4.2.2 Standard time scales

The equations of motion Equation (4.2) and the light time solution Equation (4.4) are both written in terms of an independent time variable, which is called the ephemeris time, or ET. Ephemeris time is simply coordinate time in the chosen coordinate frame, such as a solar system barycentric frame. As such, the ephemeris time differs from the standard International Atomic Time (TAI, Temps Atomique International), measured in SI (Système International) seconds relative to a given epoch, namely the beginning of the year 1958.

When a solar system barycentric frame of reference is used to integrate the equations of motion, the relationship between ET and TAI can be expressed, to an accuracy that is sufficient for the purposes of the Pioneer projectFootnote 18, as

$${\rm{ET}} - {\rm{TAI}} = (32.184 + 1.657 \times {10^{- 3}}\sin \,E)\,{\rm{seconds,}}$$


$$E = M + 0.01671\,\sin \,M,$$
$$M = 6.239996 + 1.99096871 \times {10^{- 7}}t,$$

and t is ET in seconds since the J2000 epoch (noon, January 1, 2000). For further details, including higher accuracy time conversion formulae, see the relevant literature [89, 90, 91, 92, 235, 240] (in particular, see Equations (2–26) through (2–28) in [240].)

There exist alternate expressions with up to several hundred additional periodic terms, which provide greater accuracies. The use of these extended expressions provide transformations of ET — TAI to accuracies of 1 ns [240].

For the purposes of the investigation of the Pioneer anomaly, the Station Time (ST) is especially significant. The station time is the time kept by the ultrastable oscillators of DSN stations, and it is measured in Universal Coordinated Time (UTC). All data records generated by DSN stations are timestamped using ST, that is, UTC as measured by the station’s clock.

UTC is a discontinuous time scale; it is similar to TAI, except for the regular insertion of leap seconds, which are used to account for minute variations in the Earth’s rate of rotation. Converting from UTC to international atomic time (TAI) requires the addition or subtraction of the appropriate number of leap seconds (ranging between 10 and 32 during the lifetime of the Pioneer missions.) For more details see [240, 331].

4.3 Nongravitational forces external to the spacecraft

Even in the vacuum of interplanetary space, the motion of a spacecraft is governed by more than just gravity. There are several nongravitational forces acting on a spacecraft, many of which must be taken into account in order to achieve an orbit determination accuracy at the level of the Pioneer anomaly. (For a general introduction to nongravitational forces acting on spacecraft, consult [186] and p. 125 in [210].) To determine the Pioneer orbits to sufficient precision, orbit determination programs must take into account these nongravitational accelerations unless a particular force can be demonstrated to be too small in magnitude to have a detectable effect on the spacecraft’s orbit.

In the presentation below of the standard modeling of small nongravitational forces, we generally follow the discussion in [27], starting with nongravitational forces that originate from sources external to the spacecraft, and followed by a review of forces of on-board origin. We also discuss effects acting on the radio signal sent to, or received from, the spacecraft.

4.3.1 Solar radiation pressure

Most notable among the sources for the forces external to the Pioneer spacecraft is the solar pressure. This force is a result of the exchange of momentum between solar photons and the spacecraft, as solar photons are absorbed or reflected by the spacecraft. This force can be significant in magnitude in the vicinity of the Earth, at ∼ 1 AU from the Sun, especially when considering spacecraft with a large surface area, such as those with large solar panels or antennas. For this reason, solar pressure models are usually developed before a spacecraft is launched. These models take into account the effective surface areas of the portions of the spacecraft exposed to sunlight, and their thermal and optical properties. These models offer a computation of the acceleration of the spacecraft due to solar pressure as a function of solar distance and spacecraft orientation.

The simplest way of modeling solar pressure is by using a “flat plate” model. In this case, the spacecraft is treated as a flat surface, oriented at same angle with respect to incoming solar rays. The surface absorbs some solar heat, while it reflects the rest; reflection can be specular or diffuse. A flat plate model is fully characterized by three numbers: the area of the plate, its specular and its diffuse reflectivities. This model is particularly applicable to Pioneer 10 and 11 throughout most of their mission, as the spacecraft were oriented such that their large parabolic dish antennas were aimed only a few degrees away from the Sun, and most of the spacecraft body was behind the antenna, not exposed to sunlight.

In the case of a flat plate model, the force produced by the solar pressure can be described using a combination of several force vectors. One vector, the direction of which coincides with the direction of incoming solar radiation, represents the force due to photons from solar radiation intercepted by the spacecraft. The magnitude of this vector Fintcpt is proportional to the solar constant at the spacecraft’s distance from the Sun, multiplied by the projected area of the flat plate surface:

$${{\bf{F}}_{{\rm{intcpt}}}} = {{{f_ \odot}A} \over {c{r^2}}}({\bf{n}} \cdot {\bf{k}}){\bf{n}},$$

where r is the Sun-spacecraft vector, n = r/r is the unit vector in this direction with r = ∣r∣, A is the effective area of the spacecraft (i.e., flat plate), k is a unit normal vector to the flat plate, f is the solar radiation constant at 1 AU from the Sun and c is the speed of light. The standard value of the solar radiation constant is f ≃ 1367 AU2 Wm−2 when A is measured in units of m2 and r, in units of AU. According to Equation (4.8), approximately 65 W of intercepted sunlight can produce a force comparable in magnitude to that of the Pioneer anomaly; in contrast, in the vicinity of the Earth, the Pioneer 10 and 11 spacecraft intercepted ∼ 7 kW of sunlight, indicating that solar pressure is truly significant, even as far away from the Sun as Saturn, for precision orbit determination.

Equation (4.8) reflects the amount of momentum carried by solar photons that are intercepted by the spacecraft body. However, one must also account for the amount of momentum carried away by photons that are reflected or re-emitted by the spacecraft body. These momenta depend on the material properties of the spacecraft exterior surfaces. The absorptance coefficient α determines the amount of sunlight absorbed (i.e., not reflected) by spacecraft materials. The emittance coefficient ϵ determines the efficience with which the spacecraft radiates (absorbed or internally generated) heat relative to an idealized black body. Finally, the specularity coefficient σ determines the direction in which sunlight is reflected: a fully specular surface reflects sunlight like a mirror, whereas a diffuse (Lambertian) surface reflects light in the direction of its normal. Together, these coefficients can be used in conjunction with basic vector algebra to calculate the force acting on the spacecraft due to specular reflection:

$${{\bf{F}}_{{\rm{spec}}}} = (1 - \alpha)\sigma [{{\bf{F}}_{{\rm{intcpt}}}} - 2({{\bf{F}}_{{\rm{intcpt}}}} \cdot {\bf{k}}){\bf{k}}],$$

and the force due to diffuse reflection:

$${{\bf{F}}_{{\rm{diffuse}}}} = (1 - \alpha)(1 - \sigma)\vert {{\bf{F}}_{{\rm{intcpt}}}}\vert {\bf{k}}.$$

Lastly, the force due to solar heating (i.e., re-emission of absorbed solar heat) can be computed in conjunction with the recoil force due to internally generated heat, which is discussed later in this section.

4.3.2 Solar wind

The solar wind is a stream of charged particles, primarily protons and electrons with energies of ∼ 1 keV, ejected from the upper atmosphere of the Sun. Solar wind particles intercepted by a spacecraft transfer their momentum to the spacecraft. The acceleration caused by the solar wind has the same form as Equation (4.8), with f replaced by mpυ3n, where n ≈ 5 cm−3 is the proton density at 1 AU and υ ≈ 400 km/s is the speed of the wind (electrons in the solar wind travel faster, but due to their smaller mass, their momenta are much smaller than the momenta of the protons). Thus,

$${{\bf{F}}_{{\rm{solar}}\,{\rm{wind}}}} = {{{m_p}{\upsilon ^3}nA} \over {c{r^2}}}({\bf{n}} \cdot {\bf{k}}){\bf{n}} \simeq 7 \times {10^{- 4}}{A \over {{r^2}}}({\bf{n}} \cdot {\bf{k}}){\bf{n}}.$$

Because the density can change by as much as 100%, the exact acceleration is unpredictable. Nonetheless, as confirmed by actual measurementFootnote 19, the magnitude of Equation (4.11) is at least 105 times smaller than the direct solar radiation pressure. This contribution is completely negligible [27], and therefore, it can be safely ignored.

4.3.3 Interaction with planetary environments

When a spacecraft is in the vicinity of a planetary body, it interacts with that body in a variety of ways. In addition to the planet’s gravity, the spacecraft may be subjected to radiation pressure from the planet, be slowed by drag in the planet’s extended atmosphere, and it may interact with the planet’s magnetosphere.

For instance, for Earth orbiting satellites, the Earth’s optical albedo of [235]:

$${\alpha _{{\rm{Earth}}}} \simeq 0.34$$

yields typical albedo accelerations of 10–35% of the acceleration due to solar radiation pressure. On the other hand, when the spacecraft is in the planetary shadow, it does not receive direct sunlight.

Atmospheric drag can be modeled as follows [235]:

$${\bf{\ddot r}} = - {1 \over 2}{C_D}{A \over m}\rho \vert {\bf{\dot r}}\vert {\bf{\dot r}},$$

where r is the spacecraft’s position, ṙ its velocity, A is its cross-sectional area, m its mass, ρ is the atmospheric density, and the coefficient CD has typical values between 1.5 and 3.

The Lorentz force acting on a charged object with charge q traveling through a magnetic field with field strength B at a velocity v is given by

$${\bf{F}} = q({\bf{v}} \times {\bf{B}}).$$

Considering the velocity of the Pioneer spacecraft relative to a planetary magnetic field during a planetary encounter (up to 60 km/s during Pioneer 11’s encounter with Jupiter) and a strong planetary magnetic field (up to 1 mT for Jupiter near the poles), if the spacecraft carries a net electric charge, the resulting force can be significant: up to 60 N per Coulomb of charge. In actuality, the maximum measured magnetic field by the two Pioneers at Jupiter was 113.5 µT [269]. An upper bound of 0.1 µC exists for any positive charge carried by the spacecraft [269], but a possible negative charge cannot be excluded [269] and a negative charge as high as 10−4 C cannot be ruled out [269].

The long-term accelerations of Pioneer 10 and 11, however, remain unaffected by planetary effects, due to the fact that except for brief encounters with Jupiter and Saturn, the two spacecraft traveled in deep space, far from any planetary bodies.

4.3.4 Interplanetary magnetic fields

The interplanetary magnetic field strength is less than 1 nT [27]. Considering a spacecraft velocity of 104 m/s and a charge of 10−4 C, Equation (4.14) gives a force of 10−9 N or less, with a corresponding acceleration (assuming a spacecraft mass of ∼ 250 kg) of 4 × 10−12 m/s2 or less. This value is two orders of magnitude smaller than the anomalous Pioneer acceleration of aP = (8.74 ± 1.33) × 10−10 m/s2 (see Section 5.6).

4.3.5 Drag forces

While there have been attempts to explain the anomalous acceleration as a result of a drag force induced by exotic forms of matter (see Section 6.2), no known form of matter (e.g., gas, dust particles) in interplanetary space produces a drag force of significance.

The drag force on a sail was estimated as [261]:

$${{\bf{F}}_{{\rm{sail}}}} = - {{\mathcal K}_{d}}\rho A{\upsilon ^2},$$

where υ is the spacecraft’s velocity relative to the interplanetary medium, A its cross sectional area, ρ is the density of the interplanetary medium, and \({\mathcal K_d}\) is a dimensionless coefficient that characterizes the absorptance, emittance, and transmittance of the spacecraft with respect to the interplanetary medium.

Using the Pioneer spacecraft’s 2.74 m high-gain antenna as a sail and an approximate velocity of 10−4 m/s relative to the interplanetary medium, and assuming \({\mathcal K_d}\) to be of order unity, we can estimate a drag force of

$${{\bf{F}}_{{\rm{sail}}}} \simeq - 5.9 \times {10^8}\rho,$$

with F and ρ measured in SI units. According to this result, a density of ρ ∼ 3 × 10−16 kg/m3 or higher can produce accelerations that are comparable in magnitude to the Pioneer anomaly [195].

The density ρISD of dust of interstellar origin has been measured by the Ulysses probe [189, 262] at ρISD ≲ 3 × 10−23 kg/m3. The average interplanetary dust density, which also contains orbiting dust, is believed to be almost two orders of magnitude higher according to the consensus view [262]. However, higher dust densities are conceivable.

On the other hand, if one presumes a model density, the constancy of the observed anomalous acceleration of the Pioneer spacecraft puts upper limits on the dust density. For instance, an isothermal density model ρisothr−2 yields the limit ρisoth ≲ 5 × 10−17 (20 AU/r)2 kg/m3.

4.4 Nongravitational forces of on-board origin

Perhaps the most fundamental question concerning the anomalous acceleration of Pioneer 10 and 11 is whether or not the acceleration is due to an on-board effect: i.e., is the “anomalous” acceleration simply a result of our incomplete understanding of the engineering details of the two spacecraft? Therefore, it is essential to analyze systematically any possible on-board source of acceleration that may be present.

In the broadest terms, momentum conservation dictates that in order for an on-board effect to accelerate the spacecraft, the spacecraft must eject mass or emit radiation. As no significant anomaly occurred in the Pioneer 10 and 11 missions, it is unlikely that either spacecraft lost a major component during their cruise. In any case, such an occurrence would have resulted in a one time change in the spacecraft’s velocity, not any long-term acceleration. Therefore, it is safe to consider only the emission of volatiles as a means of mass ejection. Such emissions can be intentional (as during maneuvers) or due to unintended leaks of propellant or other volatiles on board. Radiation emitted as radiative energy is produced by on-board processes.

The spin of the Pioneer spacecraft makes it possible to apply a simplified treatment of forces of on-board origin that change slowly with time. Let us denote the unit vector normal to the spacecraft’s plane of rotation (i.e., the spin axis, which we assume to remain constant in time) by s. Then, considering a force F that is a linear function of time in a co-rotating reference frame that is attached to the spacecraft, it can be described in a co-moving (nonrotating) inertial frame as

$${\bf{F}}(t) = {{\bf{F}}_{\Vert}}(t) + {{\bf{F}}_ \bot}(t) = {{\bf{F}}_{\Vert}}(t) + {\mathbb R}(\omega t) \cdot [{{\bf{F}}_ \bot}({t_0}) + {{{\bf{\dot F}}}_ \bot} \cdot (t - {t_0})],$$

where F(t) = [F(t) · s]s is the component of F(t) parallel with the spin axis and F(t) = F(t) − F(t) is the perpendicular component of F(t). The component F accelerates the spacecraft in the spin axis direction. The displacement due to the perpendicular component can be obtained by double integration with integration limits of t = t0 and t = t0 + 2πn/ω:

$$\Delta {{\bf{x}}_ \bot} = {1 \over {{\omega ^2}m}}{\mathbb R}(\omega {t_0} - \pi) \cdot {{{\bf{\dot F}}}_ \bot}\Delta t,$$

describing an arithmetic spiral around the spin axis. This spiral vanishes (i.e., the displacement of the spacecraft remains confined along the spin axis) if = 0, and even for nonzero its radius increases only linearly with time, and thus in most cases, it can be ignored safely.

4.4.1 Modeling of maneuvers

There were several hundredFootnote 20 Pioneer 10 and Pioneer 11 maneuvers during their entire missions. The modeling of maneuvers entails significant uncertainty due to several reasons. First, the duration of a thruster firing is known only approximately and may vary between maneuvers due to thermal and mechanical conditions, aging, manufacturing deficiencies in the thruster assembly, and other factors. Second, the thrust can vary as a result of changing fuel temperature and pressure. Third, imperfections in the mechanical mounting of a thruster introduce uncertainties in the thrust direction. Lastly, after a thruster has fired, leakage may occur, producing an additional, small amount of slowly decaying thrust. When combined, these effects result in a velocity change of several mm/s.

By the time Pioneer 11 reached Saturn, the behavior of its thrusters was believed to be well understood [27]. The effectively instantaneous velocity change caused by the firing of a thruster was followed by several days of decaying acceleration due to gas leakage. This acceleration was large enough to be observable in the Doppler data [270].

The Jet Propulsion Laboratory’s analysis of Pioneer orbits included either an instantaneous velocity increment at the beginning of each maneuver (instantaneous burn model) or a constant acceleration over the duration of the maneuver (finite burn model) [27]. In both cases, the burn is characterized by a single unknown parameter. The gas leak following the burn was modeled by fitting to the post-maneuver residuals a two-parameter exponential model in the form of

$$\Delta \upsilon (t) = {\upsilon _0}\exp (- t/\tau),$$

with υ0 and τ being the unknown parameters. The typical magnitude of υ0 is several mm/s, while the time constant τ is of order ∼ 10 days. Due to the spin of the spacecraft, only acceleration in the direction of the spin axis needs to be accounted for, as accelerations perpendicular to the spin axis are averaged out over several resolutions [27, 380].

4.4.2 Other sources of outgassing

Regardless of the source of a leak, the effects of outgassing on the spacecraft are governed by the rocket equation [27]:

$$a = - {\upsilon _e}{{\dot m} \over m},$$

where the dot denotes differentiation with respect to time, and υe is the exhaust velocity. The Pioneer spacecraft mass is approximately m ≃ 250 kg. For comparison, the anomalous acceleration, aP = 8.74 × 10−10 m/s2, requires an outgassing rate of ∼ 6.89 g/yr at an exhaust velocity of 1 km/s.

The exhaust velocity υe of a hot gas, according to the rocket engine nozzle equation, can be calculated as [367]Footnote 21:

$$\upsilon _e^2 = {{2kRT} \over {(k - 1){M_{{\rm{mol}}}}}}\left[ {1 - {{\left({{{{P_e}} \over {{P_i}}}} \right)}^{(k - 1)/k}}} \right],$$

where k is the isentropic expansion factor (or heat capacity ratio, k = Cp/Cυ where Cp and Cυ are the heat capacities at constant pressure and constant volume, respectively) of the exhaust gas, T is its temperature, R = 8314 JK−1 kmol−1 is the gas constant, mmol is the molecular weight of the exhaust gas in kg/kmol, Pi is its pressure at the nozzle intake, and Pe is the exhaust pressure. At room temperature, k = 1.41 for H2, k = 1.66 for He, and k = 1.40 for O2. Typical values for liquid monopropellants are 1.7 km/s < υe < 2.9 km/s.

A review of the Pioneer 10 and 11 spacecraft design reveals only three possible sources of outgassing: the propulsion system (fuel leaks), the radioisotope thermoelectric generators, and the battery.

The propulsion system carried ∼ 30 kg of hydrazine propellant and N2 pressurant. Loss of either due to a leak could produce a constant or slowly changing acceleration term. Propellant and pressurant can be lost due to a malfunction in the propulsion system, and also due to the regular operation of thruster valves, which are known to have small, persistent leaks lasting days or even weeks after each thruster firing event, as described above in Section 4.4.1. While the possibility of additional propellant leaks cannot be ruled out, in order for such leaks to be responsible for a constant acceleration like the anomalous acceleration of Pioneer 10 and 11, they would have had to be i) constant in time; ii) the same on both spacecraft; iii) not inducing any detectable changes in the spin rate or precession. Given these considerations, Anderson et al. conservatively estimate that undetected gas leaks introduce an uncertainty not greater than

$${\sigma _{{\rm{gl}}}} = \pm 0.56 \times {10^{- 10}}{\rm{m}}/{{\rm{s}}^2}.$$

Outgassing can also occur in the radioisotope thermoelectric generators as a result of alpha decay. Each kg of 238Pu produces ∼ 0.132 g of helium annually; the total amount of helium produced by the approx. 4.6 kg of radioisotope fuel on boardFootnote 22 is, therefore, 0.6 g/year. Exterior temperatures of the RTGs at no point exceeded 320 °F=433 K. According to Equation (4.21), the corresponding exhaust velocity is 2.13 km/s, resulting in an acceleration of 1.62 × 10−10 m/s2. (This is slightly larger than the corresponding estimate in [27], where the authors adopted the figures of = 0.77 g/year and υe = 1.16 km/s.) However, the circumstances required to achieve this acceleration are highly unrealistic, requiring all the helium to be expelled at maximum efficiency and in the spin axis direction. Using a more realistic (but still conservative) scenario, Anderson et al. estimate the bias and error in acceleration due to He-outgassing as

$${a_{{\rm{He}}}} = (0.15 \pm 0.16) \times {10^{- 10}}\,{\rm{m}}/{{\rm{s}}^2}.$$

Another source of possible outgassing not previously considered may be the spacecraft’s battery. According to Equation (4.21), H2 gas leaving the battery system at a temperature of 300 K can acquire an exhaust velocity of 92.6 m/s. For O2 at 300 K, the exhaust velocity is 23.2 m/s. At these velocities, an outgassing of ∼ 74 g/year of H2 or 298 g/year of O2 can produce an acceleration equal to aP, so the battery cannot be ruled out in principle as a source of a near constant acceleration term. However, no realistic construction [83] for a 5 A, 11.3 V AgCd battery would provide near enough volatile electrolites for such outgassing to occur, and in any case, the nominal performance of the battery system for a far longer time period than designed indicates that no significant loss of volatiles from the battery has taken place. A conservative (but still generous) estimate using a battery of maximum weight, 2.35 kg, assuming a loss of 10% of its mass over 30 years, and a thrust efficiency of 50% yields

$${\sigma _{{\rm{bat}}}} = \pm 0.14 \times {10^{- 10}}\,{\rm{m}}/{{\rm{s}}^2}.$$

4.4.3 Thermal recoil forces

The spacecraft carried several on-board energy sources that produced waste heat (see Section 2.4). Most notably among these are the RTGs; additional heat was produced by electrical instrumentation. Further heat sources include Radioisotope Heater Units and the propulsion system.

As the spacecraft is in an approximate thermal steady state, heat generated on board must be removed from the spacecraft [380]. In deep space, the only mechanism of heat removal is thermal radiation: the spacecraft can be said to be radiatively coupled to the cosmic background, which can be modeled by surrounding the spacecraft with a large, hollow spherical black body at the temperature of ∼ 2.7 K.

As the spacecraft emits heat in the form of thermal photons, these also carry momentum pγ, in accordance with the well known law of pγ = /c, where ν is the photon’s frequency, h is Planck’s constant, and c is the velocity of light. This results in a recoil force in the direction opposite to that of the path of the photon. For a spherically symmetric body, the net recoil force is zero. However, if the pattern of radiation is not symmetrical, the resulting anisotropy in the radiation pattern yields a net recoil force.

The magnitude of this recoil force is a subject of many factors, including the location and thermal power of heat sources, the geometry, physical configuration, and thermal properties of the spacecraft’s materials, and the radiometric properties of its external (radiating) surfaces.

Key questions concerning the thermal recoil force that have been raised during the study of the Pioneer anomaly include [164, 245, 327]:

  • How much heat from the RTGs is reflected by the spacecraft, notably the rear of its high-gain antenna, and in what direction?

  • Was there a fore-aft asymmetry in the radiation pattern of the RTGs due to differential aging?

  • How much electrical heat generated on-board was radiated through the spacecraft’s louver system?

The recoil force due to on-board generated heat that was emitted anisotropically was recognized early as a possible origin of the Pioneer anomaly. The total thermal inventory on board the Pioneer spacecraft exceeded 2 kW throughout most of their mission durations. The spacecraft were in an approximate steady state: the amount of heat generated on-board was equal to the amount of heat radiated by the spacecraft.

The mass of the Pioneer spacecraft was ∼ 250 kg. An acceleration of 8.74 × 10−10 m/s2 is equivalent to a force of ∼ 0.22 µN acting on a ∼ 250 kg object. This is the amount of recoil force produced by a 65 W collimated beam of photons. In comparison with the available thermal inventory of 2500 W, a fore-aft anisotropy of less than 3% can account for the anomalous acceleration in its entirety. Given the complex shape of the Pioneer spacecraft, it is certainly conceivable that an anisotropy of this magnitude is present in the spacecrafts’ thermal radiation pattern.

The issue of the thermal recoil force remains a subject of on-going study, as estimates of the actual magnitude of this force may require significant revision in the light of new data and new investigations [378, 379, 380, 397].

4.4.4 The radio beam recoil force

Throughout most of their missions, the Pioneer 10 and 11 spacecraft were transmitting continuously in the direction of the Earth using a highly focused microwave radio beam that was emitted by the high gain antenna (HGA; see Section 2.4.5). The recoil force due to the radio beam can be readily calculated.

A naive calculation uses the nominal value of the radio beam’s power (8 W), multiplied by the reciprocal of the velocity of light, c−1, to obtain the radio beam recoil force. This is a useful way to estimate the recoil force, but it may need refinement.

The spacecraft’s radio transmission is concentrated into a very narrow beam: signal attenuation exceeds 20 dB at only 3.75° deviation from the antenna centerline (see Figure 3.6-13 in [292]). The projected transmitter power P0 in the beam direction can be computed using the integral \({P_0} = \int {d\theta \sin \theta \mathcal P(\theta)}\) where \(\mathcal P(\theta)\) is the angular power distribution of the antenna. Given the antenna power distribution, we find that P0 = P, where P is the total power radiated by the antenna, to an accuracy much better than 1%. For this reason, the shape of the transmission beam needs not be taken into account when computing the recoil force.

However, as discussed in Section 2.4.5, the power of the spacecraft’s transmitter was not constant in time: if the telemetry readings are accepted as reliable, transmitter power may have decreased by as much as 3 W or more near the end of Pioneer 10’s mission. Furthermore, some (estimated ∼ 10%) of the radio beam may have missed the antenna dish altogether, resulting in a reduced efficiency with which the energy of the spacecraft’s transmitter is converted into momentum.

Note that the navigational model that was used to navigate the Pioneers did not include this effect. It became clear only recently leading to the need to include this model as a part of the on-going efforts to re-analyze the Pioneer data.

4.5 Effects on the radio signal

The radio signal to or from the Pioneer spacecraft travels several billion kilometers in interplanetary space. Unsurprisingly, the interplanetary environment, notably charged particles emitted by the Sun and the gravitational fields in the solar system all affect the length of the path that the radio signal travels and its frequency.

The communication antennas of the DSN complex that are used to exchange data with the Pioneer spacecraft are located on the Earth’s surface. This introduces many corrections to the modeling of the uplinked or downlinked radio signal due to the orbital motion, rotation, internal dynamics and atmosphere of our home planet.

4.5.1 Plasma in the solar corona and weighting

The interplanetary medium in the solar system is dominated by the solar wind, i.e., charged particles originating from the Sun. Although their density is low, the presence of these particles has a noticeable effect on a radio frequency signal, especially when the signal passes relatively close to the Sun.

Delay due to solar plasma is a function of the electron density in the plasma. Although this can vary significantly as a result of solar activity, the propagation delay Δt (in microseconds) can be approximated using the formula [148]:

$$\Delta t = - {1 \over {2c{n_{{\rm{crit}}}}(f)}}\int\nolimits_ \oplus ^{SC} {d\ell {n_e}(t,\,{\bf{r}}),}$$

where f is the signal frequency, ncrit(f) is is the critical plasma density for frequency f that is given by

$${n_{{\rm{crit}}}}(f) = 1.240 \times {10^4}{\left({{f \over {1\,{\rm{MHz}}}}} \right)^2}{\rm{c}}{{\rm{m}}^{- 3}},$$

and ne is the electron density as a function of time and position r, which is integrated along the propagation path between the spacecraft and the Earth.

We write the electron density as a sum of a static, steady-state part, ne(r) and fluctuation δne(t, r) [392]:

$${n_e}(t,{\bf{r}}) = {n_e}({\bf{r}}) + \delta {n_e}(t,\,{\bf{r}}).$$

The second term, which is difficult to quantify, has only a small effect on the Doppler observable [392], except at conjunction, when noise due to the solar corona dominates the Doppler observable. In contrast, the steady-state behavior of the solar corona is well known and can be approximated using the formula [19, 148, 242, 243]:

$${n_e}(t,{\bf{r}}) = A{\left({{{{R_ \odot}} \over r}} \right)^2} + B{\left({{{{R_ \odot}} \over r}} \right)^{2.7}}{e^{- {{\left[ {{\phi \over {{\phi _0}}}} \right]}^2}}} + C{\left({{{{R_ \odot}} \over r}} \right)^6}.$$

where R0 = 6.96 × 108 m is the solar radius, and r is the distance from the Sun along the propagation path.

Using Equation (4.28) in Equation (4.25), we obtain the range model [27]:

$$\Delta \,{\rm{range =}} \pm {\left({{{{f_0}} \over f}} \right)^2}\left[ {A\left({{{{R_ \odot}} \over \rho}} \right)F + B{{\left({{{{R_ \odot}} \over \rho}} \right)}^{1.7}}{e^{- {{\left[ {{\phi \over {{\phi _0}}}} \right]}^2}}} + C{{\left({{{{R_ \odot}} \over \rho}} \right)}^5}} \right],$$

where f0 = 2295 MHz is the reference frequency used for the analysis of Pioneer 10, ρ is the impact parameter with respect to the Sun, and F is a light-time correction factor, which is given for distant spacecraft as

$$F = F(\rho, {r_T},{r_E}) = {1 \over \pi}\left[ {\arctan \left({{{\sqrt {r_T^2 - {\rho ^2}}} \over \rho}} \right) + \arctan \left({{{\sqrt {r_E^2 - {\rho ^2}}} \over \rho}} \right)} \right],$$

where rT and rE are the heliocentric radial distances to the target and to the Earth, respectively. The sign of the solar corona range correction is negative for Doppler measurements (positive for range).

The values of the parameters A, B, and C are: A = 6.0 × 103, B = 2.0 × 104, C = 0.6 × 106, all in meters [27].

4.5.2 Effects of the ionosphere

As the radio signal to or from the spacecraft travels through the Earth’s ionosphere, it suffers an additional propagation delay due to the presence of charged particles. This delay Δt can be modeled as [208, 324]

$$\Delta {t_{{\rm{iono}}}} = - {1 \over {cN\,\sin \,\theta}}\int\nolimits_0^{{h_{\max}}} {{N_{{\rm{iono}}}}\,dh,}$$

where θ is the antenna elevation, N is the atmospheric refractivity index, Niono is the ionospheric refractivity index, and hmax is the height of the ionosphere. N can be approximated at 106, while Niono is well approximated by the formula

$${N_{{\rm{iono}}}} = - 40.28 \times {10^6}{{{n_e}} \over {{f^2}}},$$

where ne is the electron density and f is the signal frequency in Hz. We introduce the total electron content,

$${N_e} = \int\nolimits_0^{{h_{\max}}} {{n_e}\,dh,}$$

, which allows us to express the propagation delay in the form,

$$\Delta {t_{{\rm{iono}}}} = {{40.28} \over {c{f^2}}}{N_e},$$

with c = 3 × 108 m/s. For European latitudes, the total electron content may vary from very few electrons at night to (20−,100) × 1016 electrons during the day at various stages during the solar cycle.

4.5.3 Effects of the troposphere

Chao ([353]; see also [121, 382, 166, 205, 330]) estimates the delay due to signal propagation through the troposphere using the following formula:

$$\Delta {l_{{\rm{tropo}}}} = {1 \over {\sin \,\theta + A/(\tan \,\theta + B)}},$$

where Δltropo is the additional propagation path, θ is the elevation angle, and A = Adry + Awet and B = Bdry + Bwet are coefficients defined as

$${A_{{\rm{dry}}}} = 0.00143,$$
$${A_{{\rm{wet}}}} = 0.00035,$$
$${B_{{\rm{dry}}}} = 0.0445,$$
$${B_{{\rm{wet}}}} = 0.017.$$

Unfortunately, historical weather data going back over 30 years may not be available for most DSN stations. In the absence of such data, C.B. Markwardt suggests that seasonal weather data or historical weather data from nearby weather stations can be used to achieve good modeling accuracy.Footnote 23

4.5.4 The effect of spin

The radio signal emited by the DSN and the radio signal returned by the Pioneer 10 and 11 spacecraft are circularly polarized. The spacecraft themselves are spinning, and the spin axis coincides with the axis of the HGA. Therefore, every revolution of the spacecraft adds a cycle to both the radio signal received by, and that transmitted by the spacecraft.

At a nominal rate of 4.8 revolutions per minute, the spacecraft spin adds 0.08 Hz to the radio signal frequency in each direction.

The sign of the spin contribution to the spacecraft frequency depends on whether or not the radio signal is left or right circularly polarized, and the direction of the spacecraft’s rotation.

The rotation of the spacecraft is clockwise [292] as viewed from a direction behind the spacecraft, facing towards the Earth. This implies that the spacecraft spin would contribute to the frequency of a right circularly polarized (as seen from the transmitter) signal’s frequency with a positive sign. The assumption that the DSN signal is right circularly polarized is consistent with the explanation provided in [240]. This interpretation of the spacecraft’s spin in relation to the radio signal agrees with what one finds when comparing orbit data files with or without previously applied spin correction.

The total amount of spin correction, therefore, must be written as

$${\Delta _{{\rm{spin}}}}f = \left({1 + {{240} \over {221}}} \right){\omega \over {2\pi}},$$

where ω is the angular velocity of the spacecraft, and we accounted for the Pioneer communication system turnaround ratio of 240/221.

4.5.5 Station locations

Accurate estimation of the amount of time it takes for a signal to travel between a DSN station and a distant spacecraft, and the frequency shift due to the relative motion of these, requires precise knowledge of the position and velocity of not just the spacecraft itself, but also of any ground stations participating in the communication.

DSN transmitting and receiving stations are located on the surface of the Earth Therefore, their coordinates in a solar system barycentric frame of reference are determined primarily by the orbital motion, rotation, precession and nutation of the Earth.

In addition to these motions of the Earth, station locations also change relative to a geocentric frame of reference due to tidal effects and continental drift.

Information about station locations is readily available for stations presently in operation; however, for stations that are no longer operating, or for stations that have been relocated, it is somewhat more difficult to obtain (see Section 3.1.2).

The transformation of station coordinates from a terrestrial reference frame, such as ITRF93, to a celestial (solar system barycentric) reference frame can be readily accomplished using publicly available algorithms or software libraries, such as NASA’s SPICE libraryFootnote 24 [4].

4.6 Modeling the radiometric Doppler observable

The Pioneer spacecraft were navigated using radiometric Doppler dataFootnote 25. The Doppler observable is defined as the difference between the number of cycles received by a receiving station and the number of cycles produced by a (fixed or ramped) known reference frequency, during a specific count interval.

The expected value of the Doppler observable can be calculated accurately if the trajectory of the transmitting station (e.g., a spacecraft) and receiving station (e.g., a ground-based tracking station) are known accurately, along with information about the transmission medium along the route of the received signal.

The trajectory of the spacecraft is determined using the spacecraft’s initial position and velocity according to Section 4.1, in conjunction with a model of nongravitational forces, as detailed in Section 4.4.

The signal propagation delay due to the gravitational field of solar system can be calculated using Equation (4.4). Afterwards, from the known arrival times of the first and last cycle during a Doppler count interval, the times of their transmission can be obtained. Given the known frequency of the transmitter, one can then calculate the actual number of cycles that were transmitted during this interval. Comparing the two figures gives the difference known as the Doppler residual.

The model parameters, which include an estimate of the spacecraft’s initial state vector and other factors, can be adjusted to achieve a “best fit” between model and observation. A commonly used method to achieve a best fit is the use of a least squares estimator. The “solve-for” parameters can include orbital parameters of solar system bodies; the visible light and infrared radiometric properties of the spacecraft; or properties of the Earth’s atmosphere or the interplanetary medium.

While a spacecraft is in flight, the revised model can be used to make navigational predictions and provide guidance, as depicted in Figure 4.1. In the case of the historical Doppler data of Pioneer 10 and 11, the purpose is not to navigate a live spacecraft, but to provide a model of as large a segment of the spacecraft’s trajectory as possible, using a consistent set of parameters and minimizing the model residual. Several, independent efforts to analyze the trajectories of the two spacecraft have demonstrated that this can be accomplished using multi-year spans of data with a root-mean-square model residual of no more than a few mHz.

4.7 Orbit determination and parameter estimation

The Pioneer anomaly has been verified using a variety of independent orbit determination codes. The code that was used for the initial discovery of the anomaly, JPL’s Orbit Determination Program (ODP), is probably also the most comprehensive and best tested among these, as it is the primary code that is being used to navigate US and many international spacecraft anywhere in the solar system, especially in very deep space. ODP is a complex engineering achievement that includes many thousands of lines of code that were built during the last 50 years of space exploration. The physical models in ODP draw on fundamental principles and practices developed during decades of deep space exploration (see [138, 204, 235, 240, 348, 374]). In its core, ODP relies on a program called “Regress” that calculates the computed values of Doppler (and other) observables obtained at the tracking stations of the DSN. Regress also calculates media corrections and partial derivatives of the computed values of the observables with respect to the solve-for-parameter vector-state.

An orbit determination procedure first determines the spacecraft’s initial position and velocity in a data interval. For each data interval, we then estimate the magnitudes of the orientation maneuvers, if any. The analysis uses models that include the effects of planetary perturbations, radiation pressure, the interplanetary media, general relativity, and bias and drift in the Doppler and range (if available). Planetary coordinates and solar system masses are obtained using JPL’s Export Planetary Ephemeris DEnnn, where DE stands for the Development Ephemeris and nnn is the current number. (Earlier in the study, DE200 and DE405 were used, presently DE412 is available.)

Current versions of ODP implement computations in the J2000.0 epoch. Past versions used B1950.0. (See [361] for details on the conversion of positions and proper motions between these two epochs.)

Standard ODP modeling includes a number of solid-Earth effects, namely precession, nutation, sidereal rotation, polar motion, tidal effects, and tectonic plates drift (see discussion in [27]). Model values of the tidal deceleration, nonuniformity of rotation, polar motion, Love numbers, and Chandler wobble are obtained observationally, by means of lunar and satellite laser ranging (LLR, SLR) techniques and very long baseline interferometry (VLBI). Currently this information is provided by way of the International Celestial Reference Frame (ICRF). JPL’s Earth Orientation Parameters (EOP) is a major source contributor to the ICRF.

Since the previous analysis [24, 27], physical models for the Earth’s interior and the planetary ephemeris have greatly improved. This is due to progress in GPS, SLR, LLR and VLBI techniques, Doppler spacecraft tracking, and new radio science data processing algorithms. ODP models have been updated using these latest Earth models (adopted by the IERS) and also are using the latest planetary ephemeris. This allows for a better characterization of not only the constant part of any anomalous acceleration, but also of the annual and diurnal terms detected in the Pioneer 10 and 11 Doppler residuals [27, 274, 390].

During the last few decades, the algorithms of orbital analysis have been extended to incorporate a Kalman-filter estimation procedure that is based on the concept of “process noise” (i.e., random, nonsystematic forces, or random-walk effects). This was motivated by the need to respond to the significant improvement in observational accuracy and, therefore, to the increasing sensitivity to numerous small perturbing factors of a stochastic nature that are responsible for observational noise. This approach is well justified when one needs to make accurate predictions of the spacecraft’s future behavior using only the spacecraft’s past hardware and electronics state history as well as the dynamic environmental conditions in the distant craft’s vicinity. Modern navigational software often uses Kalman-filter estimation since it more easily allows determination of the temporal noise history than does the weighted least-squares estimation.

ODP also enables the use of batch-sequential filtering and a smoothing algorithm with process noise [27]. Though the name may imply otherwise, batch-sequential processing does not involve processing the data in batches. Instead, in this approach any small anomalous forces may be treated as stochastic parameters affecting the spacecraft trajectory. As such, these parameters are also responsible for the stochastic noise in the observational data. To characterize these noise sources, we split the data interval into a number of constant or variable size batches with respect to the stochastic parameters, and make assumptions on the possible statistical properties of the noise factors. We then estimate the mean values of the unknown parameters within the batch and their second statistical moments. (More details on this “batch-sequential algorithm with smoothing filter” in [138, 240]). ODP is permanently being updated to suit the needs of precision navigation; the progress in the estimation algorithms, programming languages, models of small forces and new navigation methods have strongly supported its recent upgrades.

There have been a number of new models developed that are needed for the analysis of tracking data from interplanetary spacecraft that are now an integral part of the latest generation of the JPL’s ODP. These include an update to the relativistic formulation of the planetary and spacecraft motion, relativistic light propagation and relevant radiometric observables (i.e., Doppler, range, VLBI, and Delta Differential One-way Ranging or ΔDOR), coordinate transformation between relativistic reference frames, and several models for nongravitational forces. Details on other models and their application for the analysis of the Pioneer anomaly are in [27, 392].

5 The Original 1995–2002 Study of the Pioneer Anomaly

The Pioneer 10 and 11 spacecraft have been described informally as the most precisely navigated deep space vehicles to date. Such precise navigation [24, 27, 260, 262, 390, 391, 392, 393] was made possible by many factors, including a conservative design (see Figure 2.3 for a design drawing of the spacecraft) that placed the spacecraft’s RTGs at the end of extended booms, providing added stability and reducing thermal effects. For attitude control, the spacecraft were spin-stabilized, requiring a minimum number of attitude correction maneuvers, further reducing navigation noise. As a result, precision navigation of the Pioneer spacecraft was possible across multi-year stretches spanning a decade or more [269].

Due in part to these excellent navigational capabilities, NASA supported a proposal to extend the Pioneer 10 and 11 missions beyond the originally planned mission durations, and use the spacecraft in an attempt to perform deep space celestial mechanics experiments, as proposed by J.D. Anderson from the Jet Propulsion Laboratory (JPL). Starting in 1979, the team led by Anderson began a systematic search for unmodeled accelerations in the trajectories of the two spacecraft. The principal aim of this investigation was the search for a hypothetical tenth planet, Planet X. Later, Pioneer 10 and 11 were used to search for trans-Neptunian objects; the superior quality of their Doppler tracking results also yielded the first ever limits on low frequency gravitational radiation [27].

The acceleration sensitivity of the Pioneer 10 and 11 spacecraft was at the level of ∼ 10−10 m/s2. At this level of sensitivity, however, a small, anomalous, apparently constant Doppler frequency drift was detected [24, 27, 390].

5.1 The early evidence for the anomaly and the original study

By 1980, when Pioneer 10 had already passed a distance of ∼ 20 AU from the Sun and the acceleration contribution from solar radiation pressure on the spacecraft had decreased to less than 4 × 10−10 m/s2, the radiometric data started to show the presence of the anomalous sunward acceleration. Figure 5.1 shows these early unmodeled accelerations of Pioneer 10 (from about 1981 to 1989) and Pioneer 11 (from 1977 to 1989).

Figure 5.1
figure 28

Early unmodeled sunward accelerations of Pioneer 10 (from about 1981 to 1989) and Pioneer 11 (from 1977 to 1989). Adapted from [27], which contained this important footnote: “Since both the gravitational and radiation pressure forces become so large close to the Sun, the anomalous contribution close to the Sun in [this figure] is meant to represent only what anomaly can be gleaned from the data, not a measurement.”

The JPL team continued to monitor the unexpected anomalous accelerations of Pioneer 10 and 11. Eventually, a proposal was made to NASA to initiate a formal study. The proposal argued that the anomaly is evident in the data of both spacecraft; that no physical model available can explain the puzzling behavior; and that, perhaps, an investigation with two independent software codes is needed to exclude the possibility of a systematic error in the navigational software. NASA supported the proposed investigation and, in 1995, the formal study was initiated at JPL and, independently, at the Aerospace Corporation, focusing solely on the acceleration anomaly detected in the radiometric Doppler data of both spacecraft Pioneer 10 and 11.

Acceleration estimates for Pioneer 10 and 11 were developed at the JPL using the Orbit Determination Program (ODP). The independent analysis performed by The Aerospace Corporation utilized a different software package called Compact High Accuracy Satellite Motion Program (CHASMP). This effort confirmed the presence of the anomalous acceleration, excluded computational systematics as a likely cause, and also indicated a possible detection of a sunward acceleration anomaly in the Galileo and Ulysses spacecrafts’ signals.

Standard navigational models account for a number of post-Newtonian perturbations in the dynamics of the planets, the Moon, and spacecraft. Models for light propagation are correct to order (υ/c)2. The equations of motion of extended celestial bodies are valid to order (υ/c)4. Non-gravitational effects, such as solar radiation pressure and precessional attitude-control maneuvers, make small contributions to the apparent acceleration we have observed. The solar radiation pressure decreases as r−2; at distances > 10–15 AU it produces an acceleration in the case of the Pioneer 10 and 11 spacecraft that is much less than 8 × 10−10 m/s2, directed away from the Sun. (The acceleration due to the solar wind is roughly a hundred times smaller than this.)

The initial results of both teams (JPL and The Aerospace Corporation) were published in 1998 [24]. The JPL group concluded that there is indeed an unmodeled acceleration, aP, towards the Sun, the magnitude of which is (8.09 ± 0.20) × 10−10 m/s2 for Pioneer 10 and (8.56 ± 0.15) × 10−10 m/s2 for Pioneer 11. The formal error is determined by the use of a five-day batch sequential filter with radial acceleration as a stochastic parameter, subject to white Gaussian noise (∼ 500 independent five-day samples of radial acceleration). No magnitude variation of aP with distance was found, within a sensitivity of 2 × 10−10 m/s2 over a range of 40 to 60 AU.

To determine whether or not the anomalous acceleration is specific to the spacecraft, an attempt was made to detect any anomalous acceleration signal in the tracking data of the Galileo and Ulysses spacecraft. It soon became clear that in the case of Galileo, the effects of solar radiation and an anomalous acceleration component cannot be separated. For Ulysses, however, a possible sunward anomalous acceleration was seen in the data, at (12 ± 3) × 10−10 m/s2. Thus, the data from the Galileo and Ulysses spacecraft yielded ambiguous results for the anomalous acceleration. Nevertheless, the analysis of data from these two additional spacecraft was useful in that it ruled out the possibility of a systematic error in the DSN Doppler system that could easily have been mistaken as a spacecraft acceleration.

The systematic error found in the Pioneer 10/11 post-fit residuals could not be eliminated by taking into account all known gravitational and nongravitational forces, both internal and external to the spacecraft. A number of potential causes have been ruled out. Continuing the search for an explanation, the authors considered the following forces and effects:

  • gravity from the Kuiper belt (see Section 5.3.5);

  • gravity from the galaxy (see Section 6.6.2);

  • spacecraft gas leaks (see Section 4.4.2);

  • errors in planetary ephemerides (see Section 6.7.1);

  • errors in accepted values of the Earth’s orientation, precession and nutation (see Section 6.7.1);

  • solar radiation pressure (see Section 4.3.1);

  • precession attitude control maneuvers (see Section 6.5.2);

  • radio beam recoil force (see Section 4.4.4);

  • anisotropic thermal radiation (see Section 4.4.3).

Other possible sources of error were considered but none found to be able to explain the puzzling behavior of the two Pioneer spacecraft.

The availability of further data (the data spanned January 1987 to July 1998) from the then-still-active Pioneer 10 spacecraft allowed the collaboration to publish a revised solution for aP. In 1999, based partially on this extended data set, they published a new estimate of the average Pioneer 10 acceleration directed towards the Sun, which was found to be ∼ 7.5 × 10−10 m/s2 [390]. The analyses used JPL’s Export Planetary Ephemeris DE200, and modeled planetary perturbations, general relativistic corrections, the Earth’s nonuniform rotation and polar rotation, and effects of radiation pressure and the interplanetary medium.

A possible systematic explanation of the anomalous residuals is nonisotropic thermal radiation. The thermal power of the spacecrafts’ radioisotope thermoelectric generators was in excess of 2500 W at launch with a half-life for the 238Pu fuel of 87.74 years, and most of this power was thermally radiated into space. The power needed to explain the anomalous acceleration is ∼ 65 W. Nonetheless, anisotropically emitted thermal radiation was not seen as a likely explanation, for two reasons: first, it was assumed, after an initial analysis of the spacecraft’s geometry, that the thermal radiation would be largely isotropic, and further, the observed acceleration did not appear to be consistent with the decay rate of the radioactive fuel.

As a result of this work, it became clear that a detailed investigation of the Pioneer anomaly was needed.

5.2 The 2002 formal solution for the anomalous acceleration

The most definitive study to date of the Pioneer anomaly [27] used Pioneer 10 data from January 3, 1987 to July 22, 1998, and Pioneer 11 data from January 5, 1987 to October 1, 1990 (at this time, Pioneer 11 lost coherent mode capability, as described in Section 2.2.7). The data were again analyzed with two independently developed software packages, JPL’s ODP and The Aerospace Corporation’s CHASMP.

Following an analysis of the anomalous spin behavior of Pioneer 10 (see Section 2.3.7, and also Figure 2.16), the Pioneer 10 data set was further divided into three intervals. Interval I contained data January 3, 1987 to July 17, 1990; Interval II, from July 17, 1990 to July 12, 1992; and Interval III, from July 12, 1992 to July 22, 1998 (Table 5.1.).

Table 5.1 Acceleration estimates (in units of 10−10 m/s2) published in [27]. Two programs (JPL’s ODP and The Aerospace Corporation’s CHASMP) were used to obtain weighted least squares (WLS) and batch-sequential filtering (BSF, 1-day batch) estimates. CHASMP could also incorporate corrections based on 10.7 cm solar flux observations, called F10.7 corrections.

Analysis of results shown in Table 5.1 let the collaboration develop their estimate for the baseline “experimental” values for Pioneer 10 and 11 [27]. They found the optimally weighted least-squares solution “experimental” number for Pioneer 10:

$$a_{\exp}^{{\rm{Pio}} 10} = (7{.}84 \pm 0{.}01) \,\, \times \,\, {10^{- 10}}\,{\rm{m}}/{{\rm{s}}^{2}}{.}$$

Similarly, the experimental value for Pioneer 11 was found to be:

$$a_{\exp}^{{\rm{Pio}} 11} = (8{.}55 \pm 0{.}02) \,\, \times \,\, {10^{- 10}}\,{\rm{m}}/{{\rm{s}}^{2}}{.}$$

The main conclusions of the 2002 study [27] can be summarized as follows:

  • The tracking data of Pioneer 10 and 11 are consistent with constant sunward acceleration;

  • The magnitude of acceleration is several times noise level;

  • No known source of error or systematic bias can account for the anomalous acceleration.

Initial announcement of the anomalous acceleration (e.g., [24, 390]) triggered many proposals that invoked various conventional physics mechanisms, all aimed at explaining the origin of the anomaly. Finding a systematic origin of the proper magnitude and behavior was the main focus of these proposals. Although the most obvious explanation would be that there is a systematic origin to the effect, perhaps generated by the spacecraft themselves from anisotropic heat rejection or propulsive gas leaks, the analysis did not find evidence for either mechanism: That is, no unambiguous, on-board systematic has been discovered.

This initial search was summarized in [27, 28], where possible contributions of various mechanisms to the final solution for aP were given. The entire error budget was subdivided in three main types of effects, namely i) effects due to sources external to the spacecraft; ii) the contribution of on-board systematics; and iii) computational systematic errors (see Table 5.2.) These three categories are detailed in the following sections.

Table 5.2 Error budget: a summary of biases and uncertainties as known in 2002 [27]. Values that are the subject of on-going study are marked by an asterisk.

5.3 Sources of systematic error external to the spacecraft

External forces can contribute to all three vector components of spacecraft acceleration (in contrast, as detailed in Section 6, forces generated on board contribute primarily along the axis of rotation). However, nonradial spacecraft accelerations are difficult to observe by the Doppler technique, which measures the velocity along the Earth-spacecraft line of sight, which approximately coincides with the spacecraft spin axis.

Following [24, 27, 390], we first consider forces that affect the spacecraft motion, such as those due to i) solar-radiation pressure, and ii) solar wind pressure. We then discuss the effects on the propagation of the radio signal that are from iii) the solar corona and its mismodeling, iv) electromagnetic Lorentz forces, v) the influence of the Kuiper belt, vi) the phase stability of the reference atomic clocks, and vii) the mechanical and phase stability of the DSN antennae, together with influence of the station locations and troposphere and ionosphere contributions. Although some of the mechanisms detailed below are near the limit for contributing to the final error budget, it was found that none of them could explain the behavior of the detected signal. Moreover, some were three orders of magnitude or more too small.

5.3.1 Direct solar radiation pressure and mass

[27] estimated the systematic error from solar radiation pressure on the Pioneer 10 spacecraft over the interval from 40 to 70.5 AU, and for Pioneer 11 from 22.4 to 31.7 AU. Using Equation (4.8) they estimated that when the spacecraft reached 10 AU, the solar radiation acceleration was 18.9 × 10−10 m/s2 decreasing to 0.39 × 10−10 m/s2 at 70 AU. Because this contribution falls off with the inverse square of the spacecraft’s heliocentric distance, it can bias the Doppler determination of a constant acceleration. By taking the average of the inverse square acceleration curve over the Pioneer distance, [27] estimated the error in the acceleration of the spacecraft due to solar radiation pressure. This error, in units of 10−10 m/s, is σsp = 0.001 for Pioneer 10 over the interval from 40 to 70.5 AU, and six times this amount for Pioneer 11 over the interval from 22.4 to 31.7 AU. In addition the uncertainty in the spacecraft’s mass for the studied data interval also introduced a bias of bsp = 0.03 × 10−10 m/s2 in the acceleration value. These estimates resulted in the error estimates

$$\delta a_{{\rm{sp}}}^{{\rm{Pio}}\,10} = {b_{{\rm{sp}}}} \pm \sigma _{{\rm{sp}}}^{{\rm{Pio}}\,10} = (0.03 \pm 0.001) \times {10^{- 10}}\,{\rm{m}}/{{\rm{s}}^2},$$
$$\delta a_{{\rm{sp}}}^{{\rm{Pio}}\,11} = {b_{{\rm{sp}}}} \pm \sigma _{{\rm{sp}}}^{{\rm{Pio}}\,11} = (0.03 \pm 0.006) \times {10^{- 10}}\,{\rm{m}}/{{\rm{s}}^2}.$$

5.3.2 The solar wind

The acceleration caused by solar wind particles intercepted by the spacecraft can be estimated, as discussed in Section 4.3.2. Due to variations with a magnitude of up to 100%, the exact acceleration is unpredictable, but its magnitude is small, therefore its contribution to the Pioneer acceleration is completely negligible. Based on these arguments, the authors of [27] concluded that the total uncertainty in aP due to solar wind can be limited as

$${\sigma _{{\rm{sw}}}} \leq {10^{- 15}}\,{\rm{m}}/{{\rm{s}}^2}.$$

5.3.3 The effects of the solar corona

Given Equation (4.30) derived in Section 4.5.1 and the values of the parameters (A, B, C) = (6.0 × 103, 2.0 × 104, 0.6 × 106), all in meters, [27] estimated the acceleration error due to the effect of the solar corona on the propagation of radio waves between the Earth and the spacecraft.

The correction to the Doppler frequency shift is obtained from Equation (4.30) by simple time differentiation. (The impact parameter depends on time as ρ = ρ(t) and may be expressed in terms of the relative velocity of the spacecraft with respect to the Earth, υ ≈ 30 km/s).

The effect of the solar corona is expected to be small on the Doppler frequency shift, which is our main observable. This is due to the fact that most of the data used for the Pioneer analysis were taken with large Sun-Earth-spacecraft angles. Further, the solar corona effect on the Doppler observable has a periodic signature, corresponding to the Earth’s orbital motion, resulting in variations in the Sun-Earth-spacecraft angle. The time-averaged effect of the corona on the propagation of the Pioneers’ radio-signals is of order

$${\sigma _{{\rm{corona}}}} = \pm 0.02 \times {10^{- 10}}\,{\rm{m}}/{{\rm{s}}^2}.$$

5.3.4 Electro-magnetic Lorentz forces

The authors of [392] considered the possibility that the Pioneer spacecraft can hold a charge and be deflected in its trajectory by Lorentz forces. They noted that this was a concern during planetary flybys due to the strength of Jupiter’s and Saturn’s magnetic fields (see Figure 2.1). The magnetic field strength in the outer solar system, ≤ 10−5 Gauss, is five orders of magnitude smaller than the magnetic field strengths measured by the spacecraft at their nearest approaches to Jupiter: 0.185 Gauss for Pioneer 10 and 1.135 Gauss for Pioneer 11. Data from the Pioneer 10 plasma analyzer can be interpreted as placing an upper bound of 0.1 µC on the positive charge during its Jupiter encounter [269].

These bounds allow us to estimate the upper limit of the contribution of the electromotive force on the motion of the Pioneer spacecraft in the outer solar system. This was accomplished in [27] using the standard formula for the Lorentz-force, F = qv × B, and found that the greatest force would be on Pioneer 11 during its closest approach to Jupiter, < 20 × 10−10 m/s2. However, once the spacecraft reached the interplanetary medium, this force would decrease to

$${\sigma _{{\rm{Lorentz}}}} \lesssim 2 \times {10^{- 14}}{\rm{m}}/{{\rm{s}}^2},$$

which is negligible.

5.3.5 The Kuiper belt’s gravity

[27] specifically studied three distributions of matter in the Kupier belt, including a uniform distribution and resonance distributions that were hypothesized in [188]. The authors assumed a total mass of one Earth mass, which is significantly larger than standard estimates. Even so, the resulting accelerations are only on the order of 10−11 m/s2, two orders of magnitude less than the observed effect. The calculated accelerations vary with time, increasing as Pioneer 10 approaches the Kuiper belt, even with a uniform density model. For these reasons, [27] excluded the dust belt as a source for the Pioneer effect.

More recent infrared observations established an upper limit of 0.3 Earth masses of Kuiper Belt dust in the trans-Neptunian region [40, 363, 371]. Therefore, for the contribution of Kuiper belt gravity, the authors of [27] placed a limit of

$${\sigma _{{\rm{KB}}}} = 0.03 \times {10^{- 10}}\,{\rm{m}}/{{\rm{s}}^2}.$$

5.3.6 Stability of the frequency references

Reliable detection of a precision Doppler observable requires a very stable frequency reference at the observing stations. High precision Pioneer 10 and 11 Doppler measurements were made using 2-way and 3-way Doppler observations. In this mode, there was no on-board frequency reference at the spacecraft; the received frequency was converted to a downlink frequency using a fixed frequency ratio (240/221), and this signal was returned to the Earth. As the round-trip light time was many hours, the stability of the frequency reference over such timescales is essential. Further, in the case of 3-way Doppler measurements when the transmitting and receiving stations were not the same, it was essential to have stable frequency references that were synchronized between ground stations of the DSN.

The stability of a clock or frequency reference is usually measured by its Allan deviation. The Allan deviation σy(τ), or its square, the Allan-variance, are defined as the variance of the frequency departure yn = 〈δν/νn (where ν is the frequency and δν is its variance during the measurement period) over a measurement period τ:

$$\sigma_y^2(\tau) = {1 \over 2}\left\langle {{{({y_{n + 1}} - {y_n})}^2}} \right\rangle,$$

where angle brackets indicate averaging.

The S-band communication systems of the DSN that were used for communicating with the Pioneer spacecraft had Allan deviations that are of order σy ∼ 1.3 × 10−12 or less for ∼ 103 s integration times [392]. Using the Pioneer S-band transmission frequency as ν ∼ 2.295 GHz, we obtain

$$\delta \nu = {\sigma _y}\nu \simeq 2.98\,{\rm{mHz}}$$

over a Doppler integration time of ∼ 103 s. Applying this figure to the case of a steady frequency drift, the corresponding acceleration error over the course of a year was estimated [27] as

$${\sigma _{{\rm{freq}}}} = 0.0003 \times {10^{- 10}}\,{\rm{m}}/{{\rm{s}}^2}.$$

5.3.7 Stability of DSN antenna complexes

The measurement of the frequency of a radio signal is affected by the stability of physical antenna structures. The large antennas of the DSN complexes are not perfectly stable. Short term effects include thermal expansion, wind loading, tides and ocean loading. Long term effects are introduced by continental drift, gravity loads and the aging of structures.

All these effects are well understood and routinely accounted for as part of DSN operations. DSN personnel regularly assess the performance of the DSN complex to ensure that operational limits are maintained [352, 353].

The authors of [27] found that none of these effects can produce a constant drift comparable to the observed Pioneer Doppler acceleration. Their analysis, which included errors due to imperfect knowledge of DSN station locations, to troposphere and ionosphere models at different stations, and to Faraday rotation effects of the atmosphere, shows a negligible contribution to the observed acceleration:

$${\sigma _{{\rm{DSN}}}} \leq {10^{- 14}}\,{\rm{m}}/{{\rm{s}}^2}.$$

5.4 Sources of systematic error internal to the spacecraft

There exist several on-board mechanisms that can contribute to the acceleration of the Pioneer spacecraft. For spinning spacecraft, like Pioneer 10 and 11, the contribution of these forces to the spacecraft’s acceleration will be primarily in the direction of the spin axis. The reason for this is that for any force that is constant in a co-rotating coordinate system, the force component that is perpendicular to the spin axis will average to zero over the course of a full revolution. Consequently, for an arbitrary force, its contribution to lateral accelerations will be limited to its time-varying component (see Section 4.4).

There are several known forces of on-board origin that can result in unmodeled accelerations. These forces, in fact, represent the most likely sources of the anomaly, in particular because previously published magnitudes of several of the considered effects are subject to revision, in view of the recently recovered telemetry data and newly developed thermal models.

On-board mechanisms that we consider in this section include: i) thruster gas leaks, ii) non-isotropic radiative cooling of the spacecraft body, iii) heat from the RTGs, iv) the radio beam reaction force, and v) the expelled helium produced within the RTG and other gas emissions.

We also review the differences in experimental results between the two spacecraft.

5.4.1 Propulsive mass expulsion

The attitude control subsystems on board Pioneer 10 and 11 were used frequently to ensure that the spacecrafts’ antennas remained oriented in the direction of the Earth. This raises the possibility that the observed anomalous acceleration is due to mismodeling of these attitude control maneuvers, or inadequate modeling of the inevitable gas leaks that occur after thruster firings.

The characteristics of propulsive gas leaks are well understood and routinely modeled by trajectory estimation software. Typical gas leaks vary in magnitude after each thruster firing, and usually decrease in time, until they become negligible.

The placement of thrusters (see Section 2.2.5) makes it highly likely that any leak would also induce unaccounted-for changes in the spacecraft’s spin and attitude.

In contrast, to produce the observed acceleration, any propulsion system leaks would have had to be i) constant in time; ii) the same on both spacecraft; iii) not inducing any detectable changes in the spin rate or precession. Given these considerations, [27] conservatively estimates that undetected gas leaks introduce an uncertainty no greater than

$${\sigma _{{\rm{gl}}}} = \pm 0.56 \times {10^{- 10}}\,{\rm{m}}/{{\rm{s}}^2}.$$

5.4.2 Heat from the RTGs

The radioisotope thermoelectric generators of the Pioneer 10 and 11 spacecraft emitted up to ∼ 2500 W of heat at the beginning of the mission, slowly decreasing to ∼ 2000 W near the end. Even a small anisotropy (< 2%) in the thermal radiation pattern of the RTGs can account, in principle, for the observed anomalous acceleration. Therefore, the possibility that the observed acceleration is due to anisotropically emitted RTG heat has been considered [27, 392].

The cylindrical RTG packages (see Section 2.2.3) have geometries that are fore-aft symmetrical. Two mechanisms were considered that would nonetheless lead to a pattern of thermal radiation with a fore-aft asymmetry.

According to one argument, heat emitted by the RTGs would be reflected anisotropically by the spacecraft itself, notably by the rear of the HGA.

[27] used the spacecraft geometry and the resultant RTG radiation pattern to estimate the contribution of the RTG heat reflecting off the spacecraft to the Pioneer anomaly. The solid angle covered by the antenna as seen from the RTG packages was estimated at ∼ 2% of 4π steradians. The equivalent fraction of RTG heat is ∼ 40 W. This estimate was further reduced after the shape of the RTGs (cylindrical with large radiating fins) and the resulting anisotropic radiation pattern of the RTGs was considered. Thus, [27] estimated that this mechanism could produce only 4 W of directed power.

The force from 4 W of directed power suggests a systematic bias of ≈ 0.55 × 10−10 m/s2. The authors also add an uncertainty of the same size, to obtain a contribution from heat reflection of

$${a_{{\rm{hr}}}} = (- 0.55 \pm 0.55) \times {10^{- 10}}\,{\rm{m}}/{{\rm{s}}^2}.$$

Another mechanism may also have contributed to a fore-aft asymmetry in the thermal radiation pattern of the RTGs. Especially during the early part of the missions, one side of the RTGs was exposed to continuous intense solar radiation, while the other side was in permanent darkness. Furthermore this side, facing deep space, was sweeping through the dust contained within the solar system. These two processes may have led to different modes of surface degradation, resulting in changing emissivities [207].

To obtain an estimate of the uncertainty, [27] considered the case when one side (fore or aft) of the RTGs has its emissivity changed by only 1% with respect to the other side.Footnote 26 In a simple cylindrical model of the RTGs, with 2000 W power (only radial emission is assumed with no loss out of the sides), the ratio of the power emitted by the two sides would be 995/1005 = 0.99, or a differential emission between the half cylinders of 10 W. Therefore, the fore/aft asymmetry toward the normal would be \(10\;{\rm{W}} \times {1 \over \pi}\;\int\nolimits_0^\pi {d\phi \sin \phi \approx 6.37\;{\rm{W}}}\). A more sophisticated model of the fin structure resulted in the slightly smaller estimate of 6.12 W, which the authors of [27] took as the uncertainty from the differential emissivity of the RTGs, to obtain an acceleration uncertainty of

$${\sigma _{{\rm{de}}}} = 0.85 \times {10^{- 10}}\,{\rm{m}}/{{\rm{s}}^2}.$$

5.4.3 Nonisotropic radiative cooling of the spacecraft

It has also been suggested that the anomalous acceleration seen in the Pioneer 10/11 spacecraft can be, “explained, at least in part, by nonisotropic radiative cooling of the spacecraft [245].” Later this idea was modified, suggesting that “most, if not all, of the unmodeled acceleration” of Pioneer 10 and 11 is due to an essentially constant supply of heat coming from the central compartment, directed out the front of the craft through the closed louvers [326].

To address the original proposal [245] and several later modifications [326, 327] and [25, 28, 325] developed a bound on the constancy of aP. This bound came from first noting the 11.5 year 1-day batch-sequential result, sensitive to time variation: aP = (7.77 ± 0.16) × 10−10 m/s2. It is conservative to take three times this error to be our systematic uncertainty for radiative cooling of the craft,

$${\sigma _{{\rm{rc}}}} = \pm 0.48 \times {10^{- 10}}\,{\rm{m}}/{{\rm{s}}^2}.$$

5.4.4 Radio beam reaction force

The emitted radio-power from the spacecraft’s HGA produces a recoil force, which is responsible for an acceleration bias, brp, on the spacecraft away from the Earth. If the spacecraft were equipped with ideal antennas, the total emitted power of the spacecrafts’ radio transmitters would be in the form of a collimated beam aimed in the direction of the Earth. In reality, the antenna is less than 100% efficient: some of the radio frequency energy from the transmitter may miss the antenna altogether, the radio beam may not be perfectly collimated, and it may not be aimed precisely in the direction of the Earth.

Therefore, using β to denote the efficiency of the antenna, we can compute an acceleration bias as

$${b_{{\rm{rp}}}} = {1 \over {mc}}\beta {P_{{\rm{rp}}}},$$

where Prp is the transmitter’s power. The nominal transmitted power of the spacecraft is 8 W. Given the m = 24 kg as the mass of a spacecraft with half its fuel gone, and using the 0.4 dB antenna error as a means to estimate the uncertainty, we obtain the acceleration figure of

$${a_{{\rm{rp}}}} = {b_{{\rm{rp}}}} \pm {\sigma _{{\rm{rp}}}} = - (1.10 \pm 0.10) \times {10^{- 10}}\,{\rm{m}}/{{\rm{s}}^2},$$

where the negative sign indicates that this acceleration is in the direction away from the Earth (and thus from the Sun), i.e., this correction actually increases the amount of anomalous acceleration required to account for the Pioneer Doppler observations [27].

5.4.5 Expelled helium produced within the RTGs

Another possible on-board systematic error is from the expulsion of the He being created in the RTGs from the α-decay of 238Pu. According to the discussion presented in Section 4.4.2, Anderson et al. estimate the bias and error in acceleration due to He-outgassing as

$${a_{{\rm{He}}}} = (0.15 \pm 0.16) \times {10^{- 10}}\,{\rm{m}}/{{\rm{s}}^2}.$$

5.4.6 Variation between determinations from the two spacecraft

Section 5.2 presented two experimental results for the Pioneer anomaly from the two spacecraft: 7.84 × 10−10 m/s2 (Pioneer 10) and 8.55 × 10−10 m/s2 (Pioneer 11). The first result represents the entire 11.5 year data period for Pioneer 10; Pioneer 11’s result represents a 3.75 year data period.

The difference between the two craft could be due to differences in gas leakage. It also could be due to heat emitted from the RTGs. In particular, the two sets of RTGs have had different histories and so might have different emissivities. Pioneer 11 spent more time in the inner solar system (absorbing radiation). Pioneer 10 has swept out more dust in deep space. Further, Pioneer 11 experienced about twice as much Jupiter/Saturn radiation as Pioneer 10.

[27] estimated the value for the Pioneer anomaly based on the two independent determinations derived from the two spacecraft, Pioneer 10 and 11. They calculated the time-weighted average of the experimental results from the two craft: [(11.5)(7.84) + (3.75)(8.55)]/(15.25) = 8.01 in units of 10−10 m/s2. This result implies a bias of b2 craft = 0.17 × 10−10 m/s2 with respect to the Pioneer 10 experimental result aP(exp) (see Equation (5.1)). We can take this number to be a measure of the uncertainty from the separate spacecraft measurements, so the overall quantitative measure is

$${a_{2\,{\rm{craft}}}} = {b_{2\,{\rm{craft}}}} \pm {\sigma _{2\,{\rm{craft}}}} = (0.17 \pm 0.17) \times {10^{- 10}}\,{\rm{m}}/{{\rm{s}}^2}.$$

5.5 Computational systematics

The third group of effects was composed of contributions from computational errors (see Table 5.2). The effects in this group dealt with i) the numerical stability of least-squares estimations, ii) accuracy of consistency/model tests, iii) mismodeling of maneuvers, and that of iv) the solar corona model used to describe the propagation of radio waves. It has also been demonstrated that the influence of v) annual/diurnal terms seen in the data on the accuracy of the estimates was small.

5.5.1 Numerical stability of least-squares estimation

The authors of [27] looked at the numerical stability of the least squares estimation algorithms and the derived solutions.

Common precision orbit determination algorithms use double precision arithmetic. The representation uses a 53-bit mantissa, equivalent to more than 15 decimal digits of precision [151]. Is this accuracy sufficient for precision orbit determination within the solar system? At solar system barycentric distances between 1 and 10 billion kilometers (1012–1013 m), 15 decimal digits of accuracy translates into a positional error of 1 cm or less. Therefore, we can conclude that double precision arithmetic is adequate in principle for modeling the orbits of Pioneer 10 and 11 in the outer solar system in a solar system barycentric reference frame. However, one must still be concerned about cumulative errors and the stability of the employed numerical algorithms.

The leading source for computational errors in finite precision arithmetic is the addition of quantities of different magnitudes, causing a loss of least significant digits in the smaller quantity. In extreme cases, this can lead to serious instabilities in numerical algorithms. Software codes that perform matrix operations are especially vulnerable to such stability issues, as are algorithms that use finite differences for solving systems of differential equations numerically.

While it is difficult to prove that a particular solution is not a result of a numerical instability, it is extremely unlikely that two independently-developed programs could produce compatible results that are nevertheless incorrect, as a result of computational error. Therefore, verifying a result using independently-developed software codes is a reliable way to exclude numerical instabilities as a possible error source, and also to put a limit on any numerical errors.

In view of the above, given the excellent agreement in various implementations of the modeling software, the authors of [27] concluded that differences in analyst choices (parameterization of clocks, data editing, modeling options, etc.) give rise to coordinate discrepancies only at the level of 0.3 cm. This number corresponds to an uncertainty in estimating the anomalous acceleration on the order of 8 × 10−14 m/s2, which was found to be negligible for the investigation.

Analysis identified, however, a slightly larger error to contend with. After processing, Doppler residuals at JPL were rounded to 15 and later to 14 significant figures. When the Block 5 receivers came online in 1995, Doppler output was further rounded to 13 significant digits. According to [27], this roundoff results in the estimate for the numerical uncertainty of

$${\sigma _{{\rm{num}}}} = \pm 0.02 \times {10^{- 10}}\,{\rm{m}}/{{\rm{s}}^2}.$$

5.5.2 Model consistency

The accuracy of navigational codes that are used to model the motion of spacecraft is limited by the accuracy of the mathematical models employed by the programs to model the solar system. The two programs used in the investigation — JPL’s ODP/Sigma modeling software and The Aerospace Corporation’s POEAS/CHASMP software package — used different parameter estimation procedures, employed different realizations of the Earth’s orientation parameters, used different planetary ephemerides, and different data editing strategies. While it is possible that some of the differences were partially masked by maneuver estimations, internal consistency checks indicated that the two solutions were consistent at the level of one part in 1015, implying an acceleration error ≤ 10−4aP [27].

The consistency of the models was verified by comparing separately the Pioneer 11 results and the Pioneer 10 results for the three intervals studied in [27]. The models differed, respectively, by (0.25, 0.21, 0.23, 0.02) m/s2. Assuming that these errors are uncorrelated, [27] computed the combined effect on anomalous acceleration aP as

$${\sigma _{{\rm{consist}}/{\rm{model}}}} = \pm 0.13 \times {10^{- 10}}\,{\rm{m}}/{{\rm{s}}^2}.$$

5.5.3 Error due to mismodeling of maneuvers

The velocity change that results from a propulsion maneuver cannot be modeled exactly. Mechanical uncertainties, fuel properties and impurities, valve performance, and other factors all contribute uncertainties. The authors of [27] found that for a typical maneuver, the standard error in the residuals is σ0 ∼ 0.095 mm/s. Given 28 maneuvers during the Pioneer 10 study period of 11.5 years, a mismodeling of this magnitude would contribute an error to the acceleration solution with a magnitude of δaman = σ0/τ = 0.07 × 10−10 m/s2. Assuming a normal distribution around zero with a standard deviation of δaman for each single maneuver, a total of N = 28 maneuvers yields a total error of

$${\sigma _{{\rm{man}}}} = {{\delta {a_{{\rm{man}}}}} \over {\sqrt N}} = 0.01 \times {10^{- 10}}\,{\rm{m}}/{{\rm{s}}^2},$$

due to maneuver mismodeling.

5.5.4 Annual/diurnal mismodeling uncertainty

In addition to the constant anomalous acceleration term, an annual sinusoid has been reported [27, 390]. The peaks of the sinusoid occur when the spacecraft is nearest to the Sun in the celestial sphere, as seen from the Earth, at times when the Doppler noise due to the solar plasma is at a maximum. A parametric fit to this oscillatory term [27, 392] modeled this sinusoid with amplitude υat = (0.1053 ± 0.0107) mm/s, angular velocity ωat = (0.0177 ± 0.0001) rad/day, and bias bat = (0.0720 ± 0.0082) mm/s, resulting in post-fit residuals of σT = 0.1 mm/s, averaged over the data interval T.

The obtained amplitude and angular velocity can be combined to form an acceleration amplitude: aat = υatωat = (0.215 ± 0.022) × 10−10 m/s2. The likely cause of this apparent acceleration is a mismodeling of the orbital inclination of the spacecraft to the ecliptic plane [27, 392].

[27] estimated the annual contribution to the error budget for aP. Combining σT and the magnitude of the annual sinusoidal term for the entire Pioneer 10 data span, they calculated

$${\sigma _{{\rm{at}}}} = 0.32 \times {10^{- 10}}\,{\rm{m}}/{{\rm{s}}^2}.$$

This number is assumed to be the systematic error from the annual term.

[27] also indicated the presence of a significant diurnal term, with a period that is approximately equal to the sidereal rotation period of the Earth, 23h56m04s.0989. The magnitude of the diurnal term is comparable to that of the annual term, but the corresponding angular velocity is much larger, resulting in large apparent accelerations relative to aP. These large accelerations, however, average out over long observational intervals, to less than 0.03 × 10−10 m/s2 over a year. The origin of the annual and diurnal terms is likely the same modeling problem [27].

These small periodic modeling errors are effectively masked by maneuvers and plasma noise. However, as they are uncorrelated with the observed anomalous acceleration (characterized by an essentially linear drift, not annual/diurnal sinusoidal signatures), they do not represent a source of systematic error.

5.6 Error budget and the final 2002 result

The results of the 2002 study [27] are summarized in Table 5.2. Sources that contribute to the overall bias and error budget are grouped depending on their origin: external to the spacecraft, generated on-board, or computational in nature. Sources of error are treated as uncorrelated; the combined error is the root sum square of the individual error values.

The contribution of effects in the first group in Table 5.2, that is, effects external to the spacecraft to the overall error budget is negligible: σexternal ∼ 0.04 × 10−10 m/s2. The second group (on-board effects) yields the largest error contribution: σon-board ∼ 1.29 × 10−10 m/s2. Lastly, computational systematics amount to σcomp ∼ 0.35 × 10−10 m/s2.

Similarly, the largest contribution to bias comes from on-board effects: σon-board ∼ 0.87 m/s2, a value that is dominated by the radio beam reaction force. External effects contribute a bias of bexternal ∼ 0.03 m/s2, while computational systematics contribute no bias.

Note that several items in Table 5.2 are marked with an asterisk, indicating that these items are the subject of an on-going new investigation of the Pioneer anomaly (discussed in Section 7).

The bias (third column) and error (fourth column) in Table 5.2 give the final acceleration result in the form

$${a_P} = {a_{P(\exp)}} + {b_P} \pm {\sigma _P},$$


$${a_{P(\exp)}} = (7.84 \pm 0.01) \times {10^{- 10}}\,{\rm{m}}/{{\rm{s}}^2}$$

is the reported formal solution for the Pioneer anomaly that was obtained with the data set available prior to 2002 [27]. Specifically, after accounting for the systematics listed in Table 5.2 and using Equations (5.25) and (5.26), the authors of [27] presented the final result of their study as

$${a_P} = (8.74 \pm 1.33) \times {10^{- 10}}\,{\rm{m}}/{{\rm{s}}^2}.$$

This 6-σ effect is clearly significant and, as of 2009, still remains unexplained.

The 2002 analysis demonstrated that after accounting for the gravitational and other large forces included in standard orbit determination programs [24, 27, 392], the anomaly in the Doppler frequency blue shift drift is uniformly changing with a rate of P = (5.99 ± 0.01) × 10−9 Hz/s [391] (see Figure 5.2). Let us denote the frequency of the signal observed by a DSN antenna as fobs, and the predicted frequency of that signal after modeling conventional forces and other signal propagation effects as fmodel. Then, for a one-way signal, the observed anomalous effect to first order in υ/c is given by fobsfmodel = −Pt. This translates to

$${\left[ {{f_{{\rm{obs}}}}(t) - {f_{{\rm{model}}}}(t)} \right]_{{\rm{DSN}}}} = - {f_0}{{{a_P}t} \over c},$$

where f0 is the DSN reference frequency [27, 391, 393] (for a discussion of the DSN sign conventions, see Endnote 38 of [27]).

Figure 5.2
figure 29

Left: Two-way Doppler residuals (observed Doppler velocity minus model Doppler velocity) for Pioneer 10. On the vertical axis, 1 Hz is equal to 65 mm/s range change per second. Right: The best fit for the Pioneer 10 Doppler residuals with the anomalous acceleration taken out. After adding one more parameter to the model (a constant radial acceleration of aP = (8.74 ± 1.33) × 10−10 m/s2) the residuals are distributed about zero Doppler velocity with a systematic variation ∼ 3.0 mm/s on a time scale of ∼ 3 months [27].

Since the publication of the 2002 study [27], many proposals have been put forth offering theoretical explanations of the anomaly. These are reviewed in the next section (Section 6). On the other hand, our knowledge of the anomaly also improved. The existence and magnitude of the anomalous acceleration has been confirmed by several independent researchers. Others have attempted to model the thermal behavior of the spacecraft, arguing that the magnitude of thermal recoil forces might have been underestimated by the 2002 study. The recovery of essentially all Pioneer 10 and 11 telemetry, as well as large quantities of archived project documentation, raised hope that it might be possible to construct a sufficiently accurate thermal model of the spacecraft using modeling software, and properly estimate the magnitude of the thermal recoil force. This remains one of several open, unresolved questions that, hopefully, will be answered in the near future as a result of on-going study, as detailed in Section 7.

6 Efforts to Explain and Study the Anomaly

Since the initial announcement of the anomalous acceleration of the Pioneer 10 and 11 spacecraft, a significant number of proposals have been made in an attempt to explain the nature of the discovered effect. The explanations targeted the effect with the properties presented in [24, 27, 390] and summarized in Section 5.6. These key properties include i) the magnitude and the apparent constancy of the anomalous acceleration, ii) its nearly Sun-pointing direction, and iii) the apparent “onset” of the anomaly. This set of “known” properties was used to analyze the mechanisms that were put forward in numerous attempts to identify the origin of the effect. Although the proposals are all very different and include conventional and new physics ideas, it is possible to place them into several broad categories. In this section we review some of these proposed mechanisms.

First, there are attempts to explain the anomaly using unmodeled conventional forces with an origin external to the spacecraft (Section 6.1), which may be both gravitational or nongravitational in nature. Some authors considered the possibility that the anomalous effect may be due to a new physics mechanism indicating, for instance, modification of gravity (Section 6.2), or may have a cosmological origin (Section 6.3). On the other hand, the fact that the Pioneer anomaly was observed in the radiometric Doppler signal opens up the possibility that the anomaly is not a dynamical effect on the trajectories of the probes but instead is due to an unmodeled effect on their radio signal (Section 6.4). We also consider proposals that attempt to explain the anomaly using unmodeled forces of on-board origin (Section 6.5). Lastly, we review miscellaneous mechanisms (Section 6.6) and some common misconceptions before moving on to a discussion of independent observational confirmations (Section 6.7) and proposals for dedicated space experiments (Section 6.8).

6.1 Unmodeled forces external to the spacecraft

The trajectory of the Pioneer spacecraft, while governed primarily by the gravity of the solar system, is nevertheless a result of a complex combination of gravitational and nongravitational forces, all of which must be taken into account for a precision orbit determination. What if some of those forces were not properly accounted for in the model, resulting in an unmodeled acceleration of the observed magnitude? Several authors considered this possibility.

6.1.1 Gravitational forces due to unknown mass distributions and the Kuiper belt

Of course, one of the most natural mechanisms to generate a putative physical force is the gravitational attraction due to a known mass distribution in the outer solar system; for instance, due to Kuiper belt objects or interplanetary dust. Anderson et al. [27] have considered such a possibility by studying various known density distributions for the Kuiper belt and concluded these density distributions are incompatible with the discovered properties of the anomaly. Even worse, these distributions cannot circumvent the constraint from the undisturbed orbits of Mars and Jupiter.

The possibility of a gravitational perturbation on the Pioneer paths has also been considered by the authors of [56, 94, 153, 253], who studied the possible effects produced by different Kuiper Belt mass distributions, and concluded that the Kuiper Belt cannot produce the observed acceleration.

Nieto [253] studied several models for 3-dimensional rings and wedges whose densities are either constant or vary as the inverse of the distance, as the inverse-squared distance, or according to the Boss-Peale model. It was demonstrated that physically viable models of this type can produce neither the magnitude nor the constancy of the Pioneer anomaly. In fact, the results emphasized the difficulty in achieving a constant acceleration within a finite cylindrically-symmetric distribution of matter. The difficulties are even stronger if one considers the amount of mass that would be needed to mimic the Pioneer anomaly.

The density of dust is not large enough to produce a gravitational acceleration on the order of aP [27, 56, 253] and also it varies greatly within the Kuiper belt, precluding any constant acceleration. In particular, Bertolami and Vieira [56] obtained the largest acceleration when the Kuiper belt was represented by a two-ring model. In this case, the following magnitude of a radial acceleration arad could be obtained (using spherical coordinates (r, θ, Φ)):

$${a_{{\rm{rad}}}}(r,\theta) = - {{GM} \over {2\pi ({R_1} + {R_2})}}\int\nolimits_0^{2\pi} {\sum\limits_{i = 1}^2 {{R_i}}} {{r - {R_i}\cos \theta \cos \phi} \over {{{({r^2} + R_i^2 - 2r\,{R_i}\cos \theta \cos \phi)}^{3/2}}}}d\phi,$$

where R1 = 39.4 AU (3:2 resonance) and R2 = 47.8 AU (2:1 resonance) are the radii of the two rings, M is the total mass of the Kuiper belt, and G is the gravitational constant. Equation (6.1) can be evaluated numerically, yielding a nonuniform acceleration that is at least an order of magnitude smaller than the Pioneer anomaly. Other dust distributions, such as those represented by a uniform disk model, a nonuniform disk model, or a toroidal model yield even smaller values. Hence, a gravitational attraction by the Kuiper belt can, to a large extent, be ruled out.

6.1.2 Drag forces due to interplanetary dust

Several nongravitational, conventional forces have been proposed by different authors to explain the anomaly. In particular, the drag force due to interplanetary dust has been investigated by the authors of [56, 253]. The acceleration adrag due to drag can be modeled as

$${a_{{\rm{drag}}}} = - {{\kappa \rho {\upsilon ^2}A} \over m},$$

where ρ is the density of the interplanetary medium, υ is the velocity of the spacecraft, A its effective cross section, m its mass, while κ is a dimensionless coefficient the value of which is 2 for reflection, 1 for absorption, and 0 for transmission of the dust particles through the spacecraft.

Using Equation (6.2) as an in situ measurement of the “apparent” density of the interplanetary medium, one obtains ρ ≃ 2.5 × 10−16 kg/m3. This is several orders of magnitude larger than the interplanetary dust density (∼ 10−21 kg/m3) reported by other spacecraft (see discussion in [262]).

The analysis of data from the inner parts of the solar system taken by the Pioneer 10 and 11 dust detectors strongly favors a spherical distribution of dust over a disk. Ulysses and Galileo measurements in the inner solar system find very few dust grains in the 10−18 − 10−12 kg range [262]. The density of dust is not large enough to produce a gravitational acceleration on the order of aP [27]. The resistance caused by the interplanetary dust is too small to provide support for the anomaly [262], so is the dust-induced frequency shift of the carrier signal.

The mechanism of drag forces due to interplanetary dust as the origin of the anomaly was discussed in detail in [262]. In particular, the authors considered this idea by taking into account the known properties of dust in the solar system, which is composed of thinly scattered matter with two main contributions:

  • Interplanetary Dust (IPD): a hot-wind plasma (mainly p and e) within the Kuiper Belt, from 30 to 100 AU with a modeled density of ρIPD ≲ 10−21 kg/m3; and

  • Interstellar Dust (ISD): composed of fractions of interstellar dust (characterized by greater impact velocity). The density of ISD was directly measured by the Ulysses spacecraft, yielding ρISD ≲ 3 × 10−23 kg/m3.

In [262] these properties were used to estimate the effect of dust on Pioneer 10 and 11 and it was found that one needs an axially-symmetric dust distribution within 20–70 AU with a constant, uniform, and unreasonably high density of ∼ 3 × 10−16 kg/m3 ≃ 3 × 105 (σIPD + σISD). Therefore, interplanetary dust cannot explain the Pioneer anomaly.

One may argue that higher densities are present within the Kuiper belt. IR observations rule out more than 0.3 Earth mass from Kuiper Belt dust in the trans-Neptunian region. Using this figure, the authors of [56] have noted that the Pioneer measurement of the interplanetary dust density is comparable to the density of various Kuiper belt models. Nonetheless, the density varies greatly within the Kuiper belt, precluding any constant acceleration.

6.2 Possibility for new physics? Modified gravity theories

Many authors investigated the possibility that the origin of the anomalous signal is “new physics” [24, 27]. This is an interesting conjecture, even though the probability is that some standard physics or some as-yet-unknown systematic will be found to explain this acceleration. Being more specific, one may ask the question, “Is it dark matter or a modification of gravity?” Unfortunately, as we discuss below, it is not easy for either of these solutions to provide a satisfactory answer.

6.2.1 Dark matter

Various distributions of dark matter in the solar system have been proposed to explain the anomaly, e.g., dark matter distributed in the form of a disk in the outer solar system with a density of ∼ 4 × 10−16 kg/m3, yielding the wanted effect. However, it would have to be a special kind of dark matter that was not seen in other nongravitational processes. Dark matter in the form of a spherical halo of a degenerate gas of heavy neutrinos around the Sun [244] and a hypothetical class of dark matter that would restore the parity symmetry, called the mirror matter [130], have also been discussed. However, it would have to be a special smooth distribution of dark matter that is not gravitationally modulated as normal matter so obviously is.

It was suggested that the observed deceleration in the Pioneer probes can be explained by the gravitational pull of a distribution of undetected dark matter in the solar system [94]. Explanations of the Pioneer anomaly involving dark matter depend on the small scale structure of Navarro-Frenk-White (NFW) haloes, which are not known. N-body simulations to investigate solar system size subhalos would require on the order of 1012 particles [246], while the largest current simulations involve around 108 particles [104]. As a consequence of this lack of knowledge about the small scale structure of dark matter, the existence of a dark matter halo around the Sun is still an open question.

It has been proposed that dark matter could become trapped in the Sun’s gravitational potential after experiencing multiple scatterings [300], perhaps combined with perturbations due to planets [86]. Moreover, the birth of the solar system itself may be a consequence of the existence of a local halo. The existence of dark matter streams crossing the solar system, perhaps forming ring-shaped caustics analogous to the dark matter ring postulated in [94], has also been considered by Sikivie [346]. Considering an NFW dark matter distribution [247], de Diego et al. [94] show that there should be several hundreds of earth masses of dark matter available in the solar system.

Gor’kavyi et al. [139] have shown that the solar system dust distributes in two dust systems and four resonant belts associated with the orbits of the giant planets. The density profile of these belts approximately follows an inverse heliocentric distance dependence law [ρ ∝ (Rk)−1, where k is a constant]. As in the case of dark matter, dust is usually modeled as a collisionless fluid as pressure, stresses, and internal friction are considered negligible. Although dust is subjected to radiation pressure, this effect is very small in the outer solar system. Gravitational pull by dark matter has also recently been considered also by Nieto [254], who also mentioned the possibility of searching for the Pioneer anomaly using the New Horizons spacecraft when the probe crosses the orbit of Saturn.

6.2.2 Modified Newtonian Dynamics (MOND)

The anomalous behavior of galaxy rotational curves led to an extensive search for dark matter particles. Some authors considered the possibility that a modification of gravity is needed to address this challenge. Consequently, there were many attempts made at constructing a theory that modifies Newton’s laws of gravity in the regime of weak gravitational fields. Presently, these efforts aim at constructing a consistent and stable theory that would also be able to account for a range of puzzling phenomena — such as flat galaxy rotational curves, gravitational lensing observations, and recent cosmological data — without postulating the existence of nonbaryonic dark matter or dark energy of yet unknown origin. Some of these novel theories were used to provide a cause of the Pioneer anomaly.

One approach to modify gravity, called Modified Newtonian Dynamics (MOND), is particularly well studied in the literature. MOND is a phenomenological modification that was proposed by Milgrom [42, 212, 211, 213, 323] to explain the “flat” rotation curves of galaxies by inducing a long-range modification of gravity. In this approach, the Newtonian force law for a test particle with mass m and acceleration a is modified as follows:

$$m{\bf{a}} = {\bf{F}} \rightarrow \mu (\vert {\bf{a}}\vert/{a_0})m{\bf{a}} = {\bf{F}},\quad {\rm{with}}\quad \mu (x) \simeq \left\{{\begin{array}{*{20}c} {1\quad {\rm{if}}\quad \vert x\vert \, \gg \,1,} \\ {x\quad {\rm{if}}\quad \vert x\vert \, \ll \,1,} \\ \end{array}} \right.$$

where μ(x) is an unspecified function (a frequent, particularly simple choice is μ(x) = x/(x + 1); other forms of μ, are also used) and a0 is some constant acceleration.

It follows from Equation (6.3) that a test particle separated by r = nr from a large mass M, instead of the standard Newtonian expression a = −GMn/r2 (which still holds when ∣a∣ ≫ a0), is subject to an acceleration that is given phenomenologically by the rule

$$\vert {\bf{a}}\vert \rightarrow \mu (\vert {\bf{a}}\vert/{a_0}){\bf{a}} \simeq \left\{{\begin{array}{*{20}c} {GM/{r^2} \propto 1/{r^2}} & {{\rm{if}}} & {a \gg {a_0}} & {({\rm{or}}\,{\rm{large}}\,{\rm{forces}}),} \\ {\sqrt {{a_0}GM}/r \propto 1/r} & {{\rm{if}}} & {a \ll {a_0}} & {({\rm{or}}\,{\rm{small}}\,{\rm{forces}}).} \\ \end{array}} \right.$$

Such a modification of Newtonian law produces a very distinct modification of galactic rotational curves. The velocities of circular orbits are modified by Equation (6.4) as

$${a_{{\rm{centrifugal}}}} = {{{\upsilon ^2}} \over r} \simeq \left\{{\begin{array}{*{20}c} {GM/{r^2}} \\ {\sqrt {{a_0}GM}/r} \\ \end{array}} \right.\quad \Rightarrow \quad {\upsilon ^2} \simeq \left\{{\begin{array}{*{20}c} {GM/r} & {{\rm{if}}} & {a \gg {a_0}} & {({\rm{or}}\,{\rm{small}}\,{\rm{distances}}),} \\ {\sqrt {{a_0}GM}} & {{\rm{if}}} & {a \ll {a_0}} & {({\rm{or}}\,{\rm{large}}\,{\rm{distances}}).} \\ \end{array}} \right.$$

With a value of a0 ≃ 1.2 × 10−10 m/s2. MOND reproduces many galactic rotation curves.

Clearly, the original MOND formulation is purely phenomenological, which drew some criticism toward the approach. However, recently a relativistic theory of gravitation that reduces to MOND in the weak-field approximation was proposed by Bekenstein in the form of the tensor-vector-scalar (TeVeS) gravity theory [41]. As the exact form of μ(x) remains unspecified in both MOND and TeVeS, it is conceivable that an appropriately chosen μ(x) might reproduce the Pioneer anomaly even as the theory’s main result, its ability to account for galaxy rotation curves, is not affected.

As far as the Pioneer anomaly is concerned, considering the strong Newtonian regime (i.e., a0GM/r2) and choosing μ(x) = 1 + ξx−1, one obtains a modification of Newtonian acceleration in the form a = −GM/r2ξa0, which reproduces the qualitative behavior implied by the observed anomalous acceleration of the Pioneers. However, Sanders [322] concludes that if the effects of a MONDian modification of gravity are not observed in the motion of the outer planets in the solar system (see Section 6.7.1 for discussion), the acceleration cannot be due to MOND. On the other hand, Bruneton and Esposito-Farèse [65] demonstrate that while it may require model choices that are not justified by underlying symmetry principles, it is possible to simultaneously account for the Pioneer anomalous acceleration and for the tests of general relativity in the solar system within a consistent field theory.

Laboratory experiments have recently reached new levels of precision in testing the proportionality of force and acceleration in Newton’s second law, F = ma, in the limit of small forces and accelerations [2, 140]. The tests were motivated to explore the acceleration scales implied by several astrophysical puzzles, such as the observed flatness of galactic rotation curves (with MOND-implied acceleration of a0 = 1.2 × 10−10 m/s), the Pioneer anomaly (with aP ∼ 9 × 10−10 m/s2) and the natural scale set by the Hubble acceleration (aH = cH ∼ 7 × 10−10 m/s2). Gundlach et al. [140] reported no violation of Newton’s second law at accelerations as small as 5 × 10−14 m/s2. The obtained result does not invalidate MOND directly as the formalism requires that the measurement must be carried out in the absence of any other large accelerations (i.e., those due to the Earth and our solar system). However, the test constrains theoretical formalisms that seek to derive MOND from fundamental principles by requiring that formalism to reproduce F = ma under laboratory conditions similar to those used in the experiment.

Finally, there were suggestions that rather then modifying laws of gravity in order to explain the Pioneer effect, perhaps we needs to modify laws of inertia instead [214]. To that extent, modified-inertia as a reaction to Unruh radiation has been considered in [201].

6.2.3 Large distance modifications of Newton’s potential

Motivated by the puzzle of the anomalous galactic rotation curves, many phenomenological models of modified Newtonian potential (leading to the changes in the gravitational inverse-square law) were considered, a Yukawa-like modification being one of the most popular scenario. Following Sanders [321], consider the ansatz:

$$U(r) = {U_{{\rm{Newton}}}}(r)\left({1 + \alpha {e^{- r/\lambda}}} \right),$$

which, as was shown in [321], is able to successfully explain many galactic rotation curves. The same expression may also be used to study the physics of the Pioneer anomaly [229]. Indeed, Equation (6.6) implies that a body moving in the gravitational field of the Sun is subject to the acceleration law:

$$a(r) = - {G_0}{{{M_ \odot}} \over {{r^2}}} + {\alpha \over {1 + \alpha}}{G_0}{{{M_ \odot}} \over {2{\lambda ^2}}} - {\alpha \over {1 + \alpha}}{G_0}{{{M_ \odot}} \over {3{\lambda ^2}}}{r \over \lambda} + \cdots,$$

where G0 = (1 + α)G is the observed gravitational constant in the limit r → 0. Identifying the second term in Equation (6.7) with the Pioneer acceleration one can solve, for instance, for the parameter α = α(aP, λ), and obtain:

$${a_P} = {\alpha \over {1 + \alpha}}{G_0}{{{M_ \odot}} \over {2{\lambda ^2}}}\quad \Rightarrow \quad \alpha = {{2{\lambda ^2}{a_P}} \over {{G_0}{M_ \odot} - 2{\lambda ^2}{a_P}}}.$$

The denominator in Equation (6.8) implies that \(\lambda > \sqrt {G{M_ \odot}/(2{a_P})}\), or λ ≥ 2.8 × 1014 m.

A combination of log λ > 16 with log ∣α∣ ≈ 0 is compatible with the existing solar system data and the Yukawa modification in the form of Equation (6.6) may provide a viable model for the Pioneer anomaly [176]. Furthermore, after rearranging the terms in Equation (6.7) as

$$a(r) = - {G_0}{{{M_ \odot}} \over {{r^2}}} + {a_P} - {2 \over 3}{a_P}{r \over \lambda} + \ldots,$$

one can note that the third term in Equation (6.9) is smaller than aP by a factor of \({2 \over 3}{r \over \lambda} \leq 0.06\) and may account for the small decrease in the observed acceleration. The values for the parameters α and λ obtained for the Pioneer anomaly are also compatible with the analysis of the galactic rotation curves. Indeed, for the case of log λ > 16 the Pioneer anomalous acceleration implies α + 1 ≤ 10−5 while the galactic curves data yields a weaker limit of α + 1 ≤ 10−1.

A modification of the gravitational field equations for a metric theory of gravity, by introducing a momentum-dependent linear relation between the Einstein tensor and the energy-momentum tensor, has been developed by Jaekel and Reynaud [156, 157, 158, 159, 160] and was shown to be able to account for ap. The authors identify two sectors, characterized by the two potentials

$${g_{00}} \simeq 1 + 2{\Phi _N},\quad {g_{00}}{g_{rr}} \simeq 1 + 2{\Phi _P},$$

where g00 and grr are components of the metric written in Eddington isotropic coordinates. The Pioneer anomaly could be accounted for by an anomaly in the Newtonian potential, δΦN ≃ (rr1)/ℓP with a characteristic length scale given by \(\ell _P^{- 1} \equiv {a_P}/{c^2}\). However, this model is likely excluded by measurements such as Viking Mars radio ranging. On the other hand, an anomaly due to the potential in the second sector in the form

$$\delta {\Phi _P} \simeq - {{{{(r - {r_1})}^2} + {\mu _P}(r - {r_1})} \over {3\kappa {\ell _P}}},$$

with κ given by 1/3κℓP ≃ 4 × 108 AU−2 and μP being a further characteristic length representing the radial derivative of the metric anomaly at the Earth’s orbit, could account for the Pioneer anomaly, and the conflict with Viking ranging data can be resolved [159].

Other related proposals include Yukawa-like or higher-order corrections to the Newtonian potential [27] and Newtonian gravity as a long wavelength excitation of a scalar condensate inducing electroweak symmetry breaking [81].

6.2.4 Scalar-tensor extensions of general relativity

There are many proposals that attempt to explain the Pioneer anomaly by invoking scalar fields. In scalar-tensor theories of gravity, the gravitational coupling strength exhibits a dependence on a scalar field φ. A general action for these theories can be written as

$$S = {{{c^3}} \over {4\pi G}}\int {{d^4}x\sqrt {- g}} \left[ {{1 \over 4}f(\varphi)R - {1 \over 2}g(\varphi){\partial _\mu}\varphi {\partial ^\mu}\varphi + V(\varphi)} \right] + \sum\limits_i {{q_i}(\varphi){{\mathcal L}_i},}$$

where f(φ), g(φ), and V(φ) are generic functions, qi(φ) are coupling functions, and \({\mathcal L_i}\) is the Lagrangian density of matter fields, as prescribed by the Standard Model of particles and fields.

Effective scalar fields are prevalent in supersymmetric field theories and string/M-theory. For example, string theory predicts that the vacuum expectation value of a scalar field, the dilaton, determines the relationship between the gauge and gravitational couplings. A general, low energy effective action for the massless modes of the dilaton can be cast as a scalar-tensor theory (as in Equation (6.12)) with a vanishing potential, where f(φ), g(φ), and qi(φ) are the dilatonic couplings to gravity, the scalar kinetic term, and the gauge and matter fields, respectively, which encode the effects of loop effects and potentially nonperturbative corrections.

Brans-Dicke theory [60] is the best known alternative scalar theory of gravity. It corresponds to the choice

$$f(\varphi) = \varphi, \quad \quad g(\varphi) = {\omega \over \varphi},\quad \quad V(\varphi) = 0.$$

In Brans-Dicke theory, the kinetic energy term of the field φ is noncanonical and the latter has a dimension of energy squared. In this theory, the constant ω marks observational deviations from general relativity, which is recovered in the limit ω → ∞. In the context of Brans-Dicke theory, one can operationally introduce Mach’s principle, which states that the inertia of bodies is due to their interaction with the matter distribution in the Universe. Indeed, in this theory the gravitational coupling is proportional to φ−1, which depends on the energy-momentum tensor of matter through the field equation □2 φ = 8π/(3 + 2ω)T where T is the trace of the matter stress-energy tensor defined as the variation of \({\mathcal L_i}\) with respect to the metric tensor.

The ω parameter can be directly related to the Eddington-Robertson (PPN) parameter γ by the relation [418]: γ = (ω + 1)/(ω + 2). The stringent observational bound resulting from the 2003 experiment with the Cassini spacecraft require that ∣ω∣ ≳ 40 000 [57, 418]. On the other hand, ω = −3/2 may be favored by cosmological observations and also offer a resolution of the Pioneer anomaly [85]. A possible resolution can be obtained by incorporating a Gauss-Bonnet term in the form of \({{\mathcal L}_{{\rm{GB}}}} = {R_{\mu \nu \rho \sigma}}{R^{\mu \nu \rho \sigma}} - 4{R_{\mu \nu}}{R^{\mu \nu}} + {R^2}\) into the Brans-Dicke version of the Lagrangian Equation (6.12) with the choice of Equation (6.13), which may allow the Eddington parameter γ to be arbitrarily close to 1, while choosing an arbitrary value for ω [10]. Another scalar-tensor model, proposed by Novati et al. [68], was also motivated in part by the observed anomalous acceleration of the two Pioneer spacecraft.

Other scalar-tensor approaches using different forms of the Lagrangian Equation (6.12) were used to investigate the anomaly. Capozziello et al. [70] developed a proposal based on flavor oscillations of neutrinos in Brans-Dicke theory; Wood [426] proposed a theory of conformal gravity with dynamical mass generation, including the Higgs scalar. Cadoni [67] studied the coupling of gravity with a scalar field with an exponential potential, while Bertolami and Paramos [53] also applied a scalar field in the context of the braneworld scenarios. In particular, Bertolami and Par ámos [53] have shown that a generic scalar field cannot explain ap; on the other hand, a non-uniformly-coupled scalar could produce the wanted effect. In addition, although braneworld models with large extra dimensions may offer a richer phenomenology than standard scalar-tensor theories, it seems difficult to find a convincing explanation for the Pioneer anomaly [54].

6.2.5 Scalar-tensor-vector modified gravity theory (MOG)

Moffat [233] attempted to explain the anomaly in the framework of Scalar-Tensor-Vector Gravity (STVG) theory. The theory originates from investigations of a nonsymmetric generalization of the metric tensor, which gives rise to a skew-symmetric field. Endowing this field with a mass led to the Metric-Skew-Tensor Gravity (MSTG) theory, while the further step of replacing the skew-symmetric field with the curl of a vector field yields STVG. The theory successfully accounts for observed galactic rotation curves, galaxy cluster mass profiles, gravitational lensing in the Bullet Cluster (1E0657-558), and cosmological observations.

The STVG Lagrangian takes the form,

$$\begin{array}{*{20}c} {{\mathcal L} = \sqrt {- g} \left\{{- {1 \over {16\pi G}}(R + 2\Lambda) - {1 \over {4\pi}}\omega \left[ {{1 \over 4}{B^{\mu \nu}}{B_{\mu \nu}} - {1 \over 2}{\mu ^2}{\phi _\mu}{\phi ^\mu} + {V_\phi}(\phi)} \right]} \right.\quad \quad \quad \quad \quad \quad \quad} \\ {\left. {- {1 \over G}\left[ {{1 \over 2}{g^{\mu \nu}}\left({{{{\nabla _\mu}G{\nabla _\nu}G} \over {{G^2}}} + {{{\nabla _\mu}\mu {\nabla _\nu}\mu} \over {{\mu ^2}}} - {\nabla _\mu}\omega {\nabla _\nu}\omega} \right) + {{{V_G}(G)} \over {{G^2}}} + {{{V_\mu}(\mu)} \over {{\mu ^2}}} + {V_\omega}(\omega)} \right]} \right\},} \\ \end{array}$$

where g is the determinant of the metric tensor, R is the Ricci-scalar, Λ is the cosmological constant, ϕν is a massive vector field with (running) mass μ, Bμν = μϕννϕμ, G is the (running) gravitational constant, ω is the (running) vector field coupling constant, and, Vϕ, VG, Vμ and Vω are the potentials associated with the vector field and the three running scalar fields.

The spherically symmetric, static vacuum solution of Equation (6.14) yields, in the weak field limit, an effective gravitational potential that is a combination of a Newtonian and a Yukawa-like term, and can be written as

$${G_{{\rm{eff}}}} = {G_N}\left({1 + \alpha - \alpha (1 + \mu r){e^{- \mu r}}} \right).$$

In earlier papers, the values of α and μ were treated as fitted parameters. This allowed Brownstein and Moffat [62, 232] to reproduce an anomalous acceleration of the correct magnitude and also account for the anomaly’s apparent “onset” at a distance of ∼ 10 AU from the Sun. More recently, the values of α and μ were derived successfully as functions of the gravitational source mass [234]. This later approach results in the prediction of Newtonian behavior within the solar system, indeed within all self-gravitating systems with a mass below several times 106 M.

6.3 Cosmologically-motivated mechanisms

There have been many attempts to explain the anomaly in terms of the expansion of the Universe, motivated by the numerical coincidence aPcH0, where c the speed of light and H0 is the Hubble constant at the present time (see Section 6.6.2 for details). These attempts were also stimulated by the fact that the initial announcement of the anomaly [24] came almost immediately after reports on the luminosity distances of type Ia supernovae [280, 310] that were followed by measurements of the angular structure of the cosmic microwave background (CMB) [93], measurements of the cosmological mass densities of large-scale structures [279] that have placed stringent constraints on the cosmological constant Λ and led to a revolutionary conclusion: The expansion of the universe is accelerating. These intriguing numerical and temporal coincidences led to heated discussion (see, e.g., [175, 273] for contrarian views) of the possible cosmological origin of the Pioneer anomaly.

Below we discuss cosmologically-motivated mechanisms used to explain the Pioneer anomaly.

6.3.1 Cosmological constant as the origin of the Pioneer anomaly

An inverse time dependence for the gravitational constant G produces effects similar to that of an expanding universe. So does a length or momentum scale-dependent cosmological term in the gravitational action functional [231, 317]. It was claimed that the anomalous acceleration could be explained in the frame of a quasi-metric theory of relativity [276]. The possible influence of the cosmological constant on the motion of inertial systems leading to an additional acceleration has been discussed [317]. Gravitational coupling resulting in an increase of the constant G with scale is analyzed by Bertolami and García-Bellido [52]. A 5-dimensional cosmological model with a variable extra dimensional scale factor in a static external space [43, 44] was also proposed. It was suggested that the coupling of a cosmological “constant” to matter [197] may provide a connection with the Pioneer anomaly.

Kagramanova et al. [163] (see also [168, 332]) have studied the effect of the cosmological constant on the outcome of the various gravitational experiments in the solar system by taking the metric of the Schwarzschild-de Sitter spacetime:

$$d{s^2} = \alpha (r)d{t^2} - \alpha {(r)^{- 1}}d{r^2} - {r^2}(d{\theta ^2} + {\sin ^2}\theta d{\phi ^2}),$$


$$\alpha (r) = 1 - {{2M} \over r} - {1 \over 3}\Lambda {r^2}$$

with Λ being the cosmological constant and M the mass of the source. (Note that Λ < 0 would result in attraction, while Λ > 0 will lead to repulsion.)

The authors of [27] have shown that if the cosmological expansion would be at the origin of the Pioneer anomaly, such a mechanism would produce an opposite sign for the effect. Taken at face value, the anomaly would imply a negative cosmological constant of Λ ∼ −3 × 10−37 m−2, which contradicts both the solar system data and the data on cosmological expansion. Indeed, the highest limit on Λ allowable by the solar system tests is set by the data on the perihelion advance, which limit the value of the cosmological constant to Λ ≤ 3 × 10−42 m−2 [332]. However, the data on the cosmological accelerated expansion yields the value of Λ ≈ 10−52 m−2, leading Kerr et al. [168] to conclude that the cosmological effects are too small to be measured in the solar system dynamical experiments.

Hackmann and Lämmerzahl [141, 142] developed the analytic solution of the geodesic equation in Schwarzschild-(anti-)de Sitter spacetimes and show that the influence of the cosmological constant on the orbits of test masses is negligible They concluded that the cosmological constant cannot be held responsible for the Pioneer anomaly

Thus, there is now a consensus that the Pioneer anomaly cannot be of a cosmological origin and, specifically, Λ cannot be responsible for the observed anomalous acceleration of the Pioneer 10 and 11 spacecraft However, the discussion is still ongoing.

6.3.2 The effect of cosmological expansion on local systems

The effect of cosmological expansion on local systems had been studied by a number of authors [21, 82, 118, 137, 202], (for reviews, see [73, 124, 177, 422]). To study the behavior of small isolated mass in expanding universe, one starts with the weak field ansatz [73, 177]:

$${g_{\mu \nu}} = {b_{\mu \nu}} + {h_{\mu \nu}},\quad {h_{\mu \nu}} \ll {b_{\mu \nu}},$$

and derives the linearized Einstein equations for hμν:

$${b^{\rho \sigma}}{D_\rho}{D_\sigma}{{\bar h}_{\mu \nu}} + {b^{\rho \sigma}}R_{\mu \,\rho \nu}^\kappa {{\bar h}_{\kappa \sigma}} = 16G\pi {T_{\mu \nu}}.$$

The relevant solution with modified Newtonian potential is given below

$${h_{00}} = {{2GM} \over R}{{\cos (\sqrt 6 \vert \dot R\vert r)} \over r} = {{2GM} \over {Rr}}\left({1 - 3{H^2}{{(Rr)}^2} + \ldots} \right).$$

The first part in Equation (6.20) is the standard Newtonian potential with the measured distance R(t)r in the denominator. Lëmmerzahl et al. [177] observed that the additional acceleration towards the gravitating body is of the second order in the Hubble constant H. As such, this potential practically does not participate in the cosmic expansion; thus, there is no support for the cosmological origin of aP.

Oliveira [273] conjectured that the solar system has escaped the gravity of the Galaxy, as evidenced by its orbital speed and radial distance and by the visible mass within the solar system radius. Spacecraft unbound to the solar system would also be unbound to the galaxy and subject to the Hubble law. However, this hypothesis produces practically unnoticeable effects.

6.3.3 The cosmological effects on planetary orbits

The cosmological effects on the planetary orbits has been addressed in many papers recently (for instance, [124]). To study the effect of cosmological modification of planetary orbit, one considers the action

$$S = - m{c^2}\int {{{\left({1 - 2{{U(x)} \over {{c^2}}} - {R^2}(t){{{{\dot x}^2}} \over {{c^2}}}} \right)}^{{1 \over 2}}}dt \approx \int {\left({- m{c^2} + mU(r) + {m \over 2}{R^2}(t)({{\dot r}^2} + {r^2}{{\dot \varphi}^2})} \right)dt.}}$$

In the case of weak time dependence of R(t) the action above has two adiabatic invariants:

$${I_\varphi} = L,\quad {I_r} = - L + {{GmM} \over R}\sqrt {{m \over {2\vert E\vert}}},$$

which determine the energy, E, momentum, p and the eccentricity, e, of an orbit:

$$E = - {{{m^3}{G^2}{M^2}} \over {2({I_r} + {I_\varphi})}},\quad p = {{I_\varphi ^2} \over {{m^2}M\,R(t)}},\quad {e^2} = 1 - {\left[ {{{{I_\varphi}} \over {{I_r} + {I_\varphi}}}} \right]^2}.$$

Lämmerzahl [176] noted that in this scenario E, e and R(t)p stay nearly constant. In addition, R(t)p and e of the planetary orbits practically do not participate in cosmic expansion. Remember that the stability of adiabatic invariants is governed by the factor of eτ/T, where τ is the characteristics time of change of external parameter (here: τ = 1/H) and T is the characteristic time of the periodic motion (here: T = periods of planets). In the the case of periodic bound orbits τ/T ∼ 1010, thus, any change of the adiabatic invariants is truly negligible.

6.3.4 Gravitationally bound systems in an expanding universe

The question of whether or not the cosmic expansion has an influence on the size of the Solar system was addressed in conjunction to the study of the Pioneer anomaly. In particular, is there a difference between locally bound and escape orbits? If the former are proven to be practically immune to cosmic expansion, what about the latter? In fact, the properties of bound (either electrically or gravitationally) systems in an expanding universe have been discussed controversially in many papers, notably by [21, 203].

Effects of cosmological expansion on local systems were addressed by a number of authors (for reviews see [73, 176]). Gautreu [137] studied the behavior of a spherical mass with the energy-stress tensor taken in the form of an ideal fluid. The obtained results show that the outer planets would tend to out-spiral away from the solar system. Anderson [21] has studied the behavior of local systems in the cosmologically-curved background. He obtained cosmological modifications of local gravitational fields with an additional drag term for escape orbits and demonstrated that Rr = const for bound orbits. Cooperstock et al. [82] used the Einstein-de Sitter universe ds2 = c2dt2R2(t)(dr2 + dΩ2) and derived a geodesic deviation equation in Fermi normal coordinates \(\ddot x - (\ddot R/R)x = 0\). Clearly, the additional terms are too small be observed in the solar system.

As far as the Pioneer anomaly is concerned, the papers above are consistent in saying that planetary orbits in the solar system should see the effects of the anomaly and aP may not be of gravitational origin. However, Anderson [21] found an interesting result that suggests that the expansion couples to escape orbits, while it does not couple to bound orbits.

To study this possibility Lämmerzahl [176] used a PPN-inspired spacetime metric:

$${g_{00}} = 1 - 2\alpha {U \over {{c^2}}} + 2\beta {{{U^2}} \over {{c^4}}},\quad {g_{ij}} = - \left({1 + 2\gamma {U \over {{c^2}}}} \right){R^2}(t){\delta _{ij}},$$

that led to the following equation of motion (written in terms of measured distances and times):

$${{{d^2}{X^i}} \over {d{T^2}}} = {{\partial U} \over {\partial {X^i}}}\left[ {\alpha + \gamma {1 \over {{c^2}}}{{\left({{{dX} \over {dT}}} \right)}^2} + (2{\alpha ^2} - \gamma \alpha - 2\beta){U \over {{c^2}}}} \right] - \left[ {\left({\alpha + \gamma} \right){1 \over {{c^2}}}{{\partial U} \over {\partial {X^j}}}{{d{X^j}} \over {dT}} + {{\dot R} \over R}} \right]{{d{X^i}} \over {dT}}.$$

Using Equation (6.25) Lämmerzahl [176] concludes that in cosmological context the behavior of bound orbits is different from that of unbound ones. However, cosmological expansion results in a decelerating drag term, which is a factor υ/c too small to account for the Pioneer effect.

6.3.5 Dark-energy-inspired f(R) gravity models

The idea that the cosmic acceleration of the Universe may be caused by a modification of gravity at very large distances, and not by a dark energy source, has recently received a great deal of attention (see [319, 351]). Such a modification could be triggered by extra space dimensions, to which gravity spreads over cosmic distances. Recently, models involving inverse powers of the curvature or other invariants have been proposed as an alternative to dark energy. Although such theories can lead to late-time acceleration, they typically result in one of two problems: either they are in conflict with tests of general relativity in the solar system, due to the existence of additional dynamical degrees of freedom [77], or they contain ghost-like degrees of freedom that seem difficult to reconcile with fundamental theories.

Gravity theories that supplement the Einstein-Hilbert Lagrangian with a nonlinear function f(R) of the curvature scalar have attracted interest in the context of inflationary cosmological scenarios, and because these theories may explain the accelerated expansion of the universe without the need to postulate scalar fields or a cosmological constant (for a review, see [351]).

As an upshot of these efforts, Bertolami et al. [50] explored a particular family of f(R) Lagrangians that gives rise to an extra acceleration in the form

$${a_E} = {{{f^2}{r^2}} \over {GM}} + 2f.$$

Assuming that fαGM/r where α is a constant leads to aEα2 = const in the limit of large r. For a galaxy, aE is naturally associated with the MOND acceleration a0, while in the solar system, it may acquire the Pioneer acceleration value, aP.

A similar result was obtained by Saffari and Rahvar [320] who used a metric formalism in f(R) modified gravity to study the dynamics of various systems from the solar system to the cosmological scales. Replacing f(R) with F(R) = df/dR, the authors postulated the ansatz F(r) = (1 + r/d)α where α ≪ 1 is a dimensionless constant and αr/d ≪ 1. They obtained an anomalous acceleration term of − α/2d, which agrees with the Pioneer anomaly for α/d ≃ 10−26 m−1.

On the other hand, Exirifard [122] demonstrated that a correction to the Einstein-Hilbert Lagrangian in the form of \(\Delta \mathcal L = {R_{\kappa \lambda \mu \nu}}{R^{\kappa \lambda \mu \nu}}\) cannot offer a covariant resolution to the Pioneer anomaly.

Capozziello et al., [72] developed a general analytic procedure to investigate the Newtonian limit of f(R) gravity. The authors discussed the Newtonian and the post-Newtonian limits of these gravity models, including the investigation of their possible relevance to the Pioneer anomaly. Capozziello et al. [69] mention that, by treating the Pioneer anomaly as a correction to the Newtonian potential, it could be studied in the general theoretical scheme of f(R) gravity theories.

6.4 Effects on the radio signal

It has been argued [301] that the cosmic expansion influences the measurement process via a change in the frequency of the traveling electromagnetic signals. However, one expects that taking all effects of the cosmic expansion on the frequency as well as on the Pioneer motion into account, the resulting acceleration is −υH0 and, thus, has the correct sign but is too small by a factor υ/c [177]. The ways in which the cosmic expansion might be responsible for aP vary considerably between the approaches. It is known [24, 27] that the very presence of the Pioneer anomalous acceleration contradicts the accurately known motion of the inner planets of our solar system. This motivated focusing on the effect of cosmic acceleration on the radio communication signal rather than on the spacecraft themselves. This mechanism might be able to overcome the apparent conflict that aP presents to modern solar system planetary ephemerides [24, 27].

6.4.1 Helicity-rotation coupling

The radio signal used for communicating with Pioneer 10 and 11 (and indeed, used routinely with other spacecraft) is circularly polarized. Therefore, the question naturally arises as to whether the coupling between the helicity of the radio signal and the rotation of either the transmitter or the receiver could contribute to the observed Doppler anomaly in the Pioneer radio signal [