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Abstract
        A previous analysis of radio Doppler and ranging data from distant Pioneer 10 and 11 spacecraft, indicated an apparent anomalous acceleration.  Several hypotheses involving new physical phenomena have been proposed to explain that apparent anomaly.  This paper shows that the anomalous acceleration of the spacecraft Pioneer 10 and 11 in the direction of the Sun is due to the presence of dust in the Kuiper belt, which has been ignored in the calculation.  These data provide the first direct measurement of dust density in the Kuiper belt, which is 1.38 x 10^-19 gr/cc.

Introduction
        A few years ago, it was observed that Pioneer 10 and 11 were subjected to anomalous constant accelerations equal to a=8 x 10^-8 cm/s², directed toward the Sun.  Consequently at a very large distance form the Sun, where the Sun’s gravity becomes negligible, the velocities of these spacecraft are constantly slowing down in their motion through outer space. 
        A huge effort has been expended looking for possible systematic effects but none has been found, as explained by Anderson et al.⁽¹⁾   The enigma is so profound that new physics has been suggested.  Crawford⁽²⁾ suggests a new gravitational redshift of the radio signal proportional to the distance to the spacecraft.  Davis⁽³⁾ considers the rest mass of the photon.  Dark matter and modified gravity is also suggested.  Rosales and Sánchez-Gomez⁽⁴⁾ propose that this is due to the local curvature in light geodesics in the expanding space-time universe.  Østvang⁽⁵⁾ claims that the gravitational field of the solar system is not static with respect to the cosmic expansion.  Belayev⁽⁶⁾ uses a compacted 5^th dimension of space to solve the problem.  Capozziello and Lambiase⁽⁷⁾ argue that this is due to the flavor oscillation of neutrinos.  Many other paranormal solutions have been claimed. 

Figure 1.   Illustration of the Pioneer 10 spacecraft.
        We demonstrate here that this anomalous acceleration can be explained using classical physics.  Calculation⁽¹⁾ implying that there is an anomalous constant acceleration directed toward the Sun, takes into account a large number of phenomena.  There has been a thorough analysis of all the possible sources of internal errors in the spacecraft, too long to be reproduced here.  As clearly explained previously by Anderson et al.⁽¹⁾, it seems that all the relevant internal problems have been solved adequately. 
Furthermore, there has also been a systematic study of the potential sources of errors, external to the spacecraft.  In order to reinvestigate these potential errors, it is necessary to take into account many phenomena explained in detail in the original paper⁽¹⁾.  Let us examine here, only these phenomena due to the environment of the spacecraft that led to the anomalous acceleration.  The complete list of those external phenomena, which have all been taken into account in the previous calculation⁽¹⁾, is:
     [1]- The pressure of the solar radiation on the spacecraft.  This pressure is due to the exchange of momentum between solar photons and the spacecraft.  This phenomenon produces an acceleration directed away from the sun. 
     [2]- The solar wind, which is the pressure of the atoms and ions emitted by the sun pushing the spacecraft away from the sun. 
     [3]- The solar corona produces a perturbation in the transmission of the radio signal between the Earth and the spacecraft.  The data obtained by radio communication needs to be analyzed in more detail. 
     [4]- In view of the fact that the spacecraft could hold an electric charge, there is a possibility of deviation of the trajectory by electromagnetic-Lorentz forces especially near Jupiter and Saturn. 
     [5]- The deflection of the spacecraft due to a gravitational perturbation due to the “mass of the Kuiper belt”. 
     [6]- The stability of the reference atomic clocks.
     [7]- The stability of the antenna together with the influence of transmission through the troposphere and ionosphere of the Earth. 
          This is the complete list of all the external forces, which have been thoroughly analyzed and in which the reader must refer in the original paper by Anderson⁽¹⁾ .  However, we show here that one important phenomenon related to the Kuiper belt [5] has been ignored.  Although the gravitational perturbation due to the Kuiper belt has been well considered, no account is taken of the momentum transfer of matter of the Kuiper belt on the spacecraft, when the spacecraft is moving through that belt. 

The Kuiper Belt.
         In 1951, astronomer Gerard Kuiper suggested that some comet-like debris from the formation of the solar system must exist beyond Neptune.  The Kuiper belt is a disk-shaped region past the orbit of Neptune roughly 30. to 100. AU from the Sun⁽⁸⁾ containing dust and many small icy bodies. It is now considered to be the source of the short-period comets.  The long-period comets are believed to be formed further away in the Oort cloud.  The understanding of that region of space is important since the study of the trans-Neptunian asteroids is a rapidly evolving field of research⁽⁹⁾, with major observational and theoretical advances in the last few years.  Similarly, the phenomenon of disk-shaped region of dust around stars is observed in several solar systems, as recorded on photographs.  For example, we see the starlight diffused on its own Kuiper belt around the star Beta Pectoris as seen in figure 2. 

Figure 2.  Photograph of Beta Pectoris surrounded by the corresponding dusty Kuiper belt.
        In fact, it is some of that dust from the Kuiper belt, which eventually is reaching the Earth neighborhood, since many tons of dust grains⁽¹⁰⁾, including samples of asteroids and comets, fall from space onto the Earth's atmosphere each day.  Once in the stratosphere this "cosmic dust" and spacecraft debris joins terrestrial particles.  Highflying aircraft with special sticky collectors can capture this dust, as it falls through the stratosphere, before it becomes mixed with Earth dust. 
        For the first time in the Pioneers 10 and 11 flights, spacecraft travel through the Kuiper belt.  Therefore at last, we have the extraordinary very first opportunity to measure “directly” the density of matter (dust and gases) in the Kuiper belt.  We examine here, whether the dust in the Kuiper belt produces a measurable effect on the spacecraft and how sensitive Pioneer 10 and 11 can detect that remnant matter.  We show here that these spacecraft are extremely sensitive to detect a minuscule amount of dust and gases.  We also show that the direct interaction of the spacecraft with the dust in the Kuiper belt, leads to a natural explanation of what appeared an anomalous acceleration by the sun. 

Calculation.
        Let us examine how the reported constant anomalous acceleration can result from the drag on the spacecraft moving through matter in space.  We show that there is a real non-gravitational acceleration of the spacecraft, resulting from the principle of momentum conservation when a moving body interacts with the stationary dust or gases in the media.  This naturally produces a slowing down of the spacecraft.  At a very large distance from the sun (~75. AU), when the Pioneer 10 and 11 spacecraft are in the Kuiper belt and move through it, we observe, as should be expected, that they move at an almost constant velocity in a direction away from the sun.  We calculate below that the reported⁽¹⁾  “anomalous acceleration”, which is an extremely slight change of velocity of the spacecraft, is due to the drag produced by matter belonging to the Kuiper belt. 
        We know that the mass of Pioneer 10 is M=241. kg. and its change of velocity⁽¹⁾ per second a (the anomalous acceleration) is 8. x 10^-8 cm/s².  Therefore, the change of momentum (Dm) of Pioneer 10 per second (which is a force) is then Dm/s=Ma=241. x 8. x 10^-10 = 1.928 x 10^-7 kg. m/s². We have m represents the meter. 
        That change of momentum of Pioneer 10 is due to the collision with dust in the Kuiper belt.  Due to its velocity, Pioneer 10 parabolic antenna, which has a radius “R” equal to 1.73 meter, sweeps a cylinder of interstellar dust with a cross section area “A” equal to A=pR².  Pioneer 10 velocity “V” is about ~12.2 Km/s.  Let us designate “d” the density of matter (dust + gas) swept by Pioneer 10 in that cylindrical volume.  We see that the mass “M” of that cylinder (of dust and gas) swept in one second is M/s=AVd.  The momentum m of a mass is defined as m=MV.  Therefore Pioneer 10 must absorb a change of momentum per second equal to Dm/s=AdV², due to the collision with the dust and gas in space.  This change of momentum is transferred to the spacecraft, which produces negative acceleration to the spacecraft (the anomalous acceleration).  Equating the change of momentum of Pioneer 10 with the change of momentum transferred by the dust gives: Dm/s=Ma= AdV².  Therefore, this shows that the dust (plus gas) density “d” of matter crossed by Pioneer 10 equals 1.38 x 10^-16 kg/m³ or 1.38 x 10^-19 gr/cc. 
        If we consider a gas, this corresponds to an extremely low gas density equal to 1.03 x 10^-16 atmosphere.  Such a density of gas is already observed in astrophysics inside some nebulae.  On the other hand, if we assume fine dust sand particles (arbitrary radius equal to ~50 microns) this density of matter in the Kuiper belt corresponds to one such a tiny grain of dust per 25000. cubic meters of space.  This amount of dust in the outer region of the solar system appears quite reasonable remembering that the daily amount of dust falling on Earth is reported as many tons of dust grains per day.  We must also notice that the real amount of matter in the Kuiper belt cannot be larger than calculated here, since it would produce a larger, non-observed drag on the Pioneer spacecraft.  We can conclude that according to the Pioneer data, we have the first direct measurement of the density of matter in the Kuiper belt in the regions crossed by Pioneer 10 and 11. 
         In a more recent paper⁽¹¹⁾, another slightly larger anomalous acceleration (12. x 10^-8 cm/s²) has also been reported using the Ulysses spacecraft data.  However, the Ulysses spacecraft is not traveling in the Kuiper belt (between 30. and 100. AU).  The Ulysses spacecraft remains much closer to the Sun between 1. and 5. AU in a region in which the gravitational field of the Sun has already been carefully tested due to the accurately known orbit of planet Mars and some asteroids.  At one AU from the sun, the interplanetary dust can even be seen “directly” from Earth when we observe the zodiacal light and the gegenschein⁽¹²⁾.  It is calculated that any anomalous gravitational acceleration larger than about (0.1 x 10^-8 cm/s²) would be measurable on Mars’s orbit.  Therefore there exists no anomalous acceleration due to gravity at that distance from the Sun. 
        Furthermore, since the “anomalous acceleration” studied here⁽¹⁾ is due to the interplanetary dust, it is normal that it has a negligible effect on Mars orbit as a consequence of the very large mass of planet Mars with respect to its cross section, when sweeping through the interplanetary dust.  In fact, we must realize that if it were due to gravity, these anomalous accelerations would also produce measurable perturbations on the orbit of planets Uranus and Pluto.  This has not been reported.  The fact that this anomalous acceleration is observed only on bodies having low masses, as on Pioneer 10 and 11, but is missing in massive bodies as Neptune and Pluto shows that its origin is not gravitational.  It is the drag due to collision with dust in the Kuiper belt, as should be expected in classical physics due to momentum conservation. 
        It is extremely interesting to note that the dust in the Kuiper belt is directly visible from Earth, using the Infrared Astronomical Satellite (IRAS)⁽¹³⁾.  This is shown on Figure 3.  The sky, as seen in several infrared wavelengths is shown in this image, assembled from the Infrared Astronomical Satellite (IRAS). The bright horizontal band (which is not of interest here) is the plane of the Milky Way.  The colors represent infrared emission detected in three of the telescope's four wavelength bands (blue is 12 microns; green is 60 microns, and red is 100 microns). Hotter material appears blue or white while the cooler material appears red. The hazy, horizontal S-shaped feature that crosses the image is faint heat emitted by dust in the plane of the solar system.   That hazy blue light is that light emitted by the dust in the Kuiper Belt.  It is the plane of that dust, which makes an angle of about 60 degrees with the Milky Way that slows down Pioneer spacecraft 10 and 11 as described above.

Figure 3.  The hazy blue feature crosses the Milky Way at about 60 degrees.   This hazy feature is due to light emitted by the heated dust of the Kuiper belt. 

        We see that the Kuiper belt is not only needed to take into account the origin of comets, but furthermore, is has been observed directly by IRAS due to the emission of light of those particles of dust heated by the Sun.  Finally, we see now in this paper, that we can measure the drag produced by that dust on the spacecraft Pioneer 10 and 11. 
        This gives a solution to the problem of the anomalous acceleration of Pioneer 10 and 11 toward the Sun, without the exceedingly improbable hypothesis of new physics^(2-7).  It is interesting to see that physics can be explained again without farfetched hypotheses.  The understanding of the origin of the anomalous acceleration, as explained here, can help NASA to plan more accurate trajectories.  Finally, considering that the Pioneer spacecraft are submitted to such an acceleration of about –8. x 10^-8 cm/s² while moving through an enormous Kuiper and Oort cloud, the spacecraft will absorb that dust due to the mechanism of accretion of dust.  After millions of years of accretion, these spacecraft will become larger and larger in time, while slowing down (unless reaccelerated later by other bodies).  Pioneer spacecraft will become the nucleus of asteroids flying away from the solar system with the interstellar dust. 
        The author acknowledges the collaboration of John Kierein for bringing up the IRAS information and also the help of G. Y. Dufour and D. O'keefe for reading and commenting the manuscript. 

References.
1 - “Study of the anomalous Acceleration of Pioneer 10 and 11”,  J. D. Anderson, P. A. Laing, E. L. Lau, A. S. Liu, M. M. Nieto, and S. G. Turyshev, Final Document  http://arXiv.org/abs/gr-qc/0104064   April 2002
2 -  D. F. Crawford, e-print:   http://arXiv.org/pdf/astro-ph/9904150 
3 -  “Pioneer 10 data yielded another fundamental physics result, a limit on the rest mass of the photon”.  See L. Davis, Jr. A. S. Goldhaber and M. M. Nieto, Phys. Rev. Lett. 35, 1402 (1975)
4 -  J. L. Rosales and J. L. Sánchez-Gomez,  Cornell, e-print   http://arXiv.org/abs/gr-qc/9810085 
5 -  D. Østvang, Cornell, e-print   http://arXiv.org/abs/gr-qc/9910054 
6 -  W. B. Belayev,  Cornell, e-print   http://arXiv.org/abs/gr-qc/9903016 
7 -  S. Capozziello and G. Lambiase, Modern Physics Lett. A. 14, 2193, (1999). Cornell, e-print  http://arXiv.org/abs/gr-qc/9910026 
8 -  Bill Arnett;  http://seds.lpl.arizona.edu/nineplanets/nineplanets/kboc.html   Oct 7, 2002, 
9 -  David Jewitt, Institute for Astronomy, http://www.ifa.hawaii.edu/faculty/jewitt/kb.html, 2680 Woodlawn Drive, Honolulu, HI 96822
10- http://www-curator.jsc.nasa.gov/curator/dust/dust.htm
11- “Indication, from Pioneer 10/11 Galileo, and Ulysses Data, of an apparent Anomalous Weak Long range Acceleration”, Physical Review Letters, John D. Anderson, Philipp A. Laing, Eunice L. Lau, Anthony S. Liu, Michael Nievo and Slava G. Tuyshev), Oct. 5, 1998
12- Zodiacal Light and the Gegenschein     http://xansrc.ee.duth.gr/html/nineplanets/medium.html
13-   http://www.ipac.caltech.edu/Outreach/Gallery/IRAS/allsky.html
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May 6,  2003
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"Anomalous Acceleration of Pioneer 10 and 11", by Paul Marmet
Paper published in:
"Proceedings of International Scientific Meetings" 
From page 334 to 337,
PIRT 2003, Physical Interpretations of Relativity, Moscow, 30 June - 03 July, 2003
Bauman Moscow State Technical University,
Physical Department & United Physical Society of Russian Federation,
Russian Gravitational Society,
British Society for the Philosophy of Science,
Liverpool University,
S.C.&T., University of Sunderland, Great Bretain,
Edited by M. C. Duffy, V. O. Gladyshev, A. N. Morozov
Moscow, Liverpool, Sunderland