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., [115Jump To The Next Citation Point116Jump To The Next Citation Point]), 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 [115116], Einstein’s general theory of relativity continues to be an active area of both theoretical and experimental research [388Jump To The Next Citation Point389Jump To The Next Citation Point]. 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 [418Jump To The Next Citation Point]. 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 [399Jump To The Next Citation Point].

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 [388Jump To The Next Citation Point389Jump To The Next Citation Point]. 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 [389Jump To The Next Citation Point399Jump To The Next Citation Point].

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 [3637]. 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 [109304366].

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 [24Jump To The Next Citation Point27Jump To The Next Citation Point390Jump To The Next Citation Point]. 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 [389Jump To The Next Citation Point]).

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 documentation, 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.

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