Pioneer 10/11 radio metric data indicate an apparent, constant skewing between predicted and observed Doppler shifts. This indicates a possible acceleration of 8.5 X 10^-8 cm/s² toward the Sun for both craft. Gravitometric models and systemic problems fail to explain the discrepancy. The anomalous signals seem to indicate an error in the relativistic Doppler equations rather than any new physics.

Anomalous Pioneer 10/11 Doppler data has been interpreted as a constant acceleration toward the Sun at ~ 8.0 X 10^-8 cm/s², [1]. Potential modeled causes change with time, act in the wrong direction or are too negligible in size to produce the offset, requiring a look at the algorithms that convert signals observed on Earth to a more inertial frame, such as the solar barycenter. Complex in practice, the modifications are conceptually simple.

The Earth is a non-inertial system compared with a reference frame stationary or linearly moving with respect to the Sun, with gravitational and motional effects modeled to great precision. Time read on an array of clocks on Earth is converted to that of a hypothetical clock in a gravitationally uniform, perfectly circular Solar orbit, referred to as TDB, Barycentric Dynamical Time. TDB represents a fixed offset from a hypothetical clock in the inertial solar barycenter reference frame, through which all observations are ultimately transformed, [2,3].

Data received from Pioneer 10/11 is transformed to TDB, then to solar barycenter time, or some similar defined-inertial frame. Data then appears as if the receivers were in the solar barycenter reference frame. The reduced data is compared to the 13 cm S-band signal transmitted to the spacecraft.

After conversion to the solar barycenter frame, two-way light times provide range data. Doppler shifted received signals used as inputs to the special relativistic Doppler expression determine the measured spacecraft velocity. Gravitational models that predict the degree of slowing over time are very precise, and provide the expected velocity for any time period or range, given the spacecraft’s trajectory history. When the observed Doppler shift is compared to the expected value from gravitational modeling, there is a constant residual offset that correlates to the numerical difference between Newtonian and special relativistic Doppler equations.

One-way special relativistic and Newtonian radial Doppler shifts, denoted "S" and "N" respectively, [4], are:

(1)

(2)

In the current study, the "time-dilation" term in (1), composed of two parts due to spacecraft and TDB velocities with respect to solar barycenter, is the primary difference between special relativistic and Newtonian radial Doppler equations.

Experiment confirms clock slowing due to an induced velocity measured against an ideal inertial frame, demonstrated notably in GPS and muon lifetime measurements at CERN. If clock slowing is due to some mechanism other than the hypothesized relativistic time dilation, then the special relativistic Doppler equation is incorrect, and the Newtonian equation is preferred, [5].

Assuming the validity of (2), then application of (1) in converting from TDB to solar barycenter frame for a 30 km/sec Earth velocity introduces an apparent constant frequency offset of -5.00 x 10^-9 Hz/Hz. An additional apparent frequency offset due to the use of (1) versus (2) in converting from solar barycenter to the 12.24 km/sec Pioneer spacecraft is -8.32 x 10^-10 Hz/Hz. Applied to the Doppler equations, these offsets translate to a residual shift in frequency of -5.83 x 10^-9 Hz/Hz, one-way only. If equation (2) is correct, these offsets will appear as a steady frequency drift in the Deep Space Network of –5.83 x 10^-9 Hz/s. Dividing by the S-band carrier frequency results in a perceived constant clock acceleration of -a_t = –2.53 x 10^-18 s/s².

Since such a consistent, systemic acceleration of all clocks is unlikely, the offset may be viewed as an anomalous spacecraft acceleration of a_p = a_tc, or a_p = 7.59 x 10^-8 cm/s², independent of distance and constant for a given spacecraft velocity. Apparent sunward acceleration increases proportionally with spacecraft velocity, not 1/r² as a gravitational force would indicate. If the anomaly is due to a preference for the Newtonian radial Doppler expression, one also expects the observed correlation between Aerospace Corporation’s Compact High Accuracy Satellite Motion Program and Jet Propulsion Laboratory’s Orbit Determination Program, as they apply the same Doppler methodology.

The authors of [1] state "it is interesting to speculate on the possibility that the origin of the anomalous signal is new physics." More likely the result is an artifact of the equations chosen to model the Doppler effect, requiring a closer look at equations comparing light-times, clock rates and Doppler frequency shifts.

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