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.

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
shift^{16}

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 , 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.

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 “-” ( is the standard deviation) test, where is the choice of the analyst (usually 4 – 10) the analyst can delete those points that lie outside the - 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.

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- 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

where is the basic standard deviation (in Hz) and 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 , where 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, . (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.

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.

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