Abstract—Radio metric data from the Pioneer 10 and 11 spacecraft indicate an apparent, constant skewing between the predicted and observed Doppler shifts. This offset has been attributed to a possible acceleration of 8.0 x 10^-8 cm/s² directed toward the sun for both craft. Any potential gravitometric models and systemic problems seem to fail in explaining this discrepancy. The value of the observed anomalous shift is shown to equal the difference between the calculated values for Newtonian and special relativistic Doppler expressions. The anomalous signals seem to indicate a preference for the Newtonian values and a deficiency of the relativistic Doppler corrections rather than any new gravitational physics.

Analysis of the two-way S-band Doppler shift in signals coherently returned from the Pioneer 10 and 11 spacecraft indicates a constant offset from the expected value. The discrepancy has been interpreted as a constant, anomalous acceleration in the direction of the Sun at ~ 8.0 X 10^-8 cm/s², independent of distance, [1]. The analysis takes account of "the effects of planetary perturbations, radiation pressure, the interplanetary media, general relativity, and bias and drift in the range and Doppler." All such effects change with time, act in the wrong direction or are too negligible in size to account for the observed frequency offsets.

Lack of physical explanation for the effect requires a close look at the algorithms used to convert a series of observed signals on the rotating, solar-orbiting Earth to a more inertial frame, such as the solar barycenter.

When the data are analyzed using Newtonian expressions for radial Doppler in place of the special relativistic expressions, the anomalies disappear. The author has published several papers that indicate a preference for the Newtonian Doppler equations. The Pioneer 10 and 11 data tend to confirm this preference, indicating limits on the applicable domain of special relativity.

While complex in practice, the algorithms used to convert Earth based observations to solar barycenter are quite simple in concept. The earth is a clearly non-inertial system (compared with a reference frame stationary or linearly moving with respect to the Sun), undergoing rotation, elliptical motion, and travel through varying gravitational fields. It is also a binary system, consisting of the moon as its companion, and thus its position varies around the center of mass of that system. Additional disturbances arise from the motion of the sun about the center of mass of the solar system, and the varying gravitational fields traversed by a signal as it crosses the paths of the large planets at various times and distances.

All of these effects are modeled to a very high precision. One can translate from the time read on a given atomic clock, or array of atomic clocks (such as the Deep Space Network), on Earth to the time of a hypothetical clock in a perfectly circular orbit and uniform gravitational field around the Sun. The time on this hypothetical clock is referred to as TDB, Barycentric Dynamical Time. TDB represents a fixed offset from a hypothetical clock at the solar barycenter, the frame through which all observations are ultimately transformed. The solar barycenter is considered to represent an inertial reference frame, [2,3]. The process and offset are illustrated in Figure 1.

Figure 1 Comparison of Earth, TDB and Solar Barycenter Time.

Data received from Pioneer 10 and 11 is transformed to TDB, then to solar barycenter time, or some similar defined-inertial frame. The data then appear as if the receivers were in the same frame of reference as the solar barycenter.

The reduced data is compared to the signal initially transmitted to the spacecraft, in this case a 13 cm S-band radio signal. The two-way light times converted to the solar barycenter frame provide range data (as described below). The light time calculations do not use the relativistic Doppler equations. The Doppler shifted received signal, also converted to the solar barycenter frame, determines the apparent speed of the craft. Solving the special relativistic Doppler expression for velocity with transmitted and received signals as inputs produces the measured spacecraft velocity. These calculations use the relativistic Doppler equations twice. Once to convert the signal received on Earth to its frequency had it been received at solar barycenter, and again to determine the frequency shift received by the spacecraft due to its velocity with respect to solar barycenter. The calculations used to determine the degree of slowing anticipated over time are very precise, and are used to calculate an anticipated velocity for any time period or range, given the history of the spacecraft trajectory and its measured range. The spacecraft are expected to be slowing as they leave the solar system, due to the pull of the sun’s gravity. When the observed Doppler shift is compared to the anticipated velocity of the craft, there is a constant residual offset in the amount of deceleration equal to a_p ~ 8.0 x 10^-8 cm/s².

The one-way special relativistic and Newtonian radial Doppler shift equations are provided below, with subscripts "S" and "N" respectively, [4].

(1)

(2)

In (1), the denominator explicitly contains the "time-dilation" term of the radial Doppler shift, which in the current study is composed of two parts, one due to TDB velocity with respect to solar barycenter, and the other due to spacecraft velocity with respect to solar barycenter. In each equation, v_r represents the radial component of the velocity between source and observer, while g depends on the total velocity. The primary difference between the special relativistic equation for radial Doppler shift and the Newtonian equation is the time-dilation term.

Experiment confirms that clocks placed in motion slow according to the relativistic time-dilation expression due to a change in the magnitude of their velocity measured against an ideal inertial frame. This is demonstrated most notably in the GPS constellation and in muon lifetimes measured at CERN. These tests do not confirm special relativistic time dilation, but they do confirm the slowing of internal and external timing mechanisms due to an imparted velocity [5]. They also lend validity to the concept of transforming TDB time to solar barycenter equivalent time.

The only way to conclusively confirm relativistic time dilation between reference frames would be to have two identical clocks constructed in their relatively moving inertial reference frames. However, this test has never been performed. For one thing, finding one suitable frame to call inertial is difficult. Finding two such frames is even harder. Secondly, in all experiments to date, all clocks in any experiment have been created and calibrated in the same reference frame, with one or more of these clocks then placed in motion to become the "moving" clock. Applying a non-inertial change in energy to a system to change its reference frame is substantially different than studying a system residing permanently in a particular inertial reference frame. This is notably expressed in the so-called twin paradox.

The Pioneer 10 and 11 spacecraft represent an excellent test of relativistic time dilation. The spacecraft are moving at near uniform speeds in a highly inertial mode. The transformations from TDB (Earth) time to inertial solar barycenter time have taken account of almost every detectable perturbation. By studying the Doppler shift in reflected signals rather than the time on clocks, the problem of accelerating one clock out of its rest frame does not arise.

If relativistic time-dilation is real, then the special relativistic Doppler equations should correctly identify the velocity of the Pioneer spacecraft. The predicted spacecraft velocity based on gravitational models, solar wind and all other factors should match the value obtained by analysis of the received signal. If the slowing of clocks is due to some mechanism other than relativistic time dilation, then the Newtonian radial Doppler equation should be preferred over the special relativistic equation, and should more correctly identify the velocity of the Pioneer spacecraft.

It is important to note that the velocity of the spacecraft is not observed directly. The velocity is obtained only after applying the two-step conversion on the received Doppler signal as described in section 2. If the Doppler equations are wrong, the velocity obtained from analysis of the received signal will be incorrect. This is not the same as saying the spacecraft is slowing down, though with no other information, it would be impossible to distinguish between erroneous algorithms, varying spacecraft velocity, or varying clock rates on Earth. The author has proposed a Galilean invariant form of Maxwell’s equations from which can be derived the Newtonian expression for radial Doppler, [6]. Thus, while compensating time for the induced slowing of clocks in solar orbit is required, the "time-dilation" offset contained in the relativistic Doppler equations would be superfluous and detrimental.

The two-way light-time range data is highly dependent on the readings of the clocks on Earth. The two-way transit times are converted from the actual Deep Space Network (DSN) clock times to TDB. To obtain a consistent range value from a common reference frame, these two-way light times are converted to the equivalent two-way time that would be seen at solar barycenter. Range is then specified consistently as the distance from solar barycenter, based on the light-times that would be seen by an ideal, inertial clock at that location. The actual procedure for determining range data on the Pioneer spacecraft is to cross correlate a phase modulated signal with a ground duplicate and inferring the time delay. The ranging data are therefore independent of the Doppler data, and, for conceptual convenience, are referred to as two-way light-times.

Special relativity also predicts a slowing of the clocks on Pioneer 10 due to its velocity with respect to solar barycenter. This presumed time differential does not affect the two-way light-times, dependent on the speed of light rather than the frequency, as the signal is effectively just reflected off the spacecraft and not processed in any significant manner. The alleged time-dilation offset does, however, affect the relativistic Doppler shift in the reflected signal. Whatever frequency is received, it is assumed to have experienced the time-dilation effects of equation (1) due to the spacecraft motion, and the presumed "measured" velocity of the craft is determined by solving the relativistic Doppler equation for velocity. Inherent in these equations is the time offset from TDB to solar barycenter as well. In converting from a received carrier frequency to an implied spacecraft velocity, both time offsets are included and considered to affect the relativistic Doppler shift.

For the calculated shift from TDB to solar barycenter, we consider only the constant velocity offset between TDB and solar barycenter time. Equation (1) in this case has a fixed offset from equation (2), independent of spacecraft velocity. Defining the velocity difference between these two reference frames as v_e, the velocity of the Earth with respect to solar barycenter, the difference in calculated Doppler shift between the Newtonian equation and that of special relativity at solar barycenter is approximately:

(3)

The data reduction algorithms also account for the Doppler effect experienced between the Pioneer 10 and 11 spacecraft and the solar barycenter due to their radial velocity with respect to the solar barycenter frame, defined as v_r. Taking the difference of equations (2) and (1) provides an expression for the difference in calculated Doppler shift for this velocity as follows:

(4)

In (3) and (4), the last two approximations hold since any effects as small as v³/c³ are well below the error limits of observation. Comparing the differences between the two Doppler approaches, Newtonian and special relativistic, with observed experimental data determines a preference for one formula over the other.

To a good first approximation, the differences calculated in (3) and (4) may be added together to obtain an estimate of the overall difference in results between the two approaches. In practice, the combination of the two velocities in the special relativistic approach is more complicated, but results in a slightly larger overall value than that indicated in this paper. The value obtained by a more thorough treatment is closer to the anomaly observed by researches than indicated in this paper, but such a technical approach does not lend itself to the conceptual treatment being presented herein.

The positive values of Equations (3) and (4) demonstrate that the expected Doppler shifted frequency using the Newtonian approach would be bluer than the expected frequency using the special relativistic approach for the multiple frame transformations used in the Pioneer 10 and 11 data reduction algorithms. If Newtonian equation (2) is correct and special relativistic equation (1) is used for data reduction, the spacecraft will appear to be slowing down at a constant rate proportional to the assumed special relativistic time-dilation offsets in the radial Doppler.

If we assume the validity of the Newtonian expression (2) for radial Doppler shift, we can calculate explicitly the errors introduced by using the special relativistic equation (1) instead. Specifically, the time-dilation offset will provide a constant skewing of the received signal for a given combination of spacecraft velocity and Earth velocity.

The first error will be in transforming the received signal from TDB (Earth time) to solar barycenter time. This transformation converts time read on the Deep Space Network of clocks to a hypothetical time on a clock at solar barycenter. The Newtonian versus special relativistic approaches in converting from TDB to solar barycenter for a 30 km/sec Earth velocity, equation (3), introduce an apparent anomalous Doppler shift of 5.00 x 10^-9 Hz/Hz.

The second error introduced would be in the assumed Doppler offset at the spacecraft itself, as compared with the solar barycenter frame. The conversion from solar barycenter to the Pioneer spacecraft, with a velocity of 12.24 km/sec, equation (4), introduces an additional apparent anomalous Doppler shift of 8.32 x 10^-10 Hz/Hz. When combined, these inferred "time-dilation" Doppler offsets translate to a residual frequency shift of 5.83 x 10^-9 Hz/Hz, one-way only.

A bluer than expected received frequency can be interpreted as due to an anomalous acceleration of tracking clocks in the Deep Space Network. In the current case, the anomalous Doppler would appear as a steady frequency drift in this clock array of –5.83 x 10 ^–9 Hz/s. The Pioneer 10 and 11 spacecraft use a 13 cm carrier frequency (2.3 x 10⁹ Hz.). This residual clock frequency drift, when divided by the carrier frequency results in a perceived clock acceleration, a_t, given in (5):

(5)

It is highly unlikely, if not impossible, that all clocks in the Deep Space Network could conspire to change in unison so uniformly as to produce this effect, though the possibility has not been ruled out.

The researchers in [1] have suggested this possibility, but one must consider what is actually being proposed. When hypothesizing an anomalous clock acceleration, such a proposal is equivalent to the following statement, "The change in the Deep Space Network of clocks is coincidentally constant, uniform across all clocks, and equal to the special relativistic time dilation term as applied to the velocity of the Earth and an artificial satellite compared to solar barycenter." The velocity of the Pioneer spacecraft have nothing to do with operating and calibrating the DSN clocks. In fact, nothing in the entire Pioneer program in anyway affects the DSN clocks. One cannot appreciate the odds against the possibility that the clocks of the DSN would just happen to be changing their rate in a manner that exactly matches the expected, yet apparently absent, time dilation offsets in Doppler algorithms between Pioneer, TDB and solar barycenter, especially since the Pioneer spacecraft are the only satellites for which such a test has ever been conclusively performed.

Alternatively, the difference between the expected and received frequencies, which matches the difference between Newtonian and special relativistic Doppler equations, has been interpreted by researchers as a spacecraft acceleration toward the sun (direction determined by the negative result), a_p, defined as in (6):

(6)

In (6), as with the presumed clock acceleration, a_p is independent of distance and is constant for a given spacecraft velocity. This is consistent with the results obtained in reference [1]. The apparent acceleration toward the Sun increases proportionally to an increase in spacecraft velocity, not proportionally to 1/r² as one might anticipate with a gravitational force. This is exactly the type of anomalous acceleration proposed for Pioneer 10 and 11 by the researchers in [1], and the value in (6), derived with round number velocity estimates for Earth and spacecraft and simplified analysis, is very close to the observed value.

There is no way to account for an apparent acceleration that increases with spacecraft velocity as an acceleration of the DSN clocks. Thus testing for the effect on a different satellite, say Ulysses, would rule out the clock acceleration possibility. Inconclusive calculation using the Ulysses satellite have cast doubt on the clock acceleration hypothesis, since the anomalous acceleration observed by that craft is on the order of 12 x 10^-8 cm/s². However, this anomaly is consistent with the analysis presented in this paper, given the trajectory and velocity of that craft.

If the anomalous signal is due to a preference for the Newtonian radial Doppler expression over that of special relativity, then a correlation between Aerospace Corporation’s Compact High Accuracy Satellite Motion Program (CHASMP) and Jet Propulsion Laboratory’s (JPL) Orbit Determination Program (ODP) would be expected, as they rely on the same Doppler methodology. This is, in fact, the case. When one is more careful to include all the "relativistic" velocity addition effects, the result of (6) becomes –8.01 x 10^-8 cm/s², even closer to the published results. The only way to determine if the observed effect is truly just an artifact of the data reduction algorithms is to look at the programs themselves, and systematically address each occurrence of special relativistic versus Newtonian Doppler calculations, actual or inherent.

The authors of [1] state that "it is interesting to speculate on the possibility that the origin of the anomalous signal is new physics." Even the title of their paper implies an "Anomalous, Weak, Long-Range Acceleration." From the above, it appears more likely that the result is "old physics," and requires a closer look at the equations used for comparing light-times, clock rates and Doppler frequency shifts.

[1] John D. Anderson, P. A. Laing, E. U. Lau, A. S. Liu, M. M. Nieto, and S. G. Turyshev, "Indication, from Pioneer 10/11, Galileo, and Ulysses Data, of an Apparent Anomalous, Weak, Long-Range Acceleration," Phys. Rev. Lett., 81, 2858 (1998).

[2] D. C. Backer, in: Timing Neutron Stars, Boston, Kluwer Academic Publishers, 1989, p. 3.

[3] J. H. Taylor, in: Timing Neutron Stars, Boston, Kluwer Academic Publishers, 1989, p. 17

[4] Richard P. Feynman, Lectures on Physics, Vol I, Menlo Park, CA, Addison-Wesley, 1965, p. 34-7.

[5] Curt Renshaw, "Moving Clocks, Reference Frames and the Twin Paradox," IEEE: Aerospace and Electronic Systems Magazine, Vol. 11, No. 1 (1996)

[6] Curt Renshaw. "The Radiation Continuum Model of Light and the Galilean Invariance of Maxwell’s Equations," IEEE: Aerospace and Electronic Systems Magazine, Vol. 13, No. 10 (1998)