### Doppler Measurement Error

The Doppler tracking of the craft is carried out in exactly the same way as described in Detection of the Pioneer Anomaly in Doppler Data.  The frequency of a returned signal from a craft is compared to that of the frequency of the sent signal.  If the craft is moving relative to the ground station then the frequency of the returned signal will be different to that sent.  If the difference is not what is expected then some unknown acceleration could be changing the velocity of the craft.

This method has associated errors which are inherent to the system and fundamentally limit the accuracy of Doppler measurements.  Two of the main errors are presented below in detail.  To summarise: if a craft is observed for 1000 seconds the two sources of errors studied in this section contribute to an error of 0.0006 mm/s (X-band) on the Doppler measurement.  This error comes about due to the choices of the project design.  Choices to do with observation time, radio signal strength and noise levels.

Which for an unknown acceleration the size of the Pioneer Anomaly provides an observed Anomaly Doppler velocity of 0.0009±0.0006 mm/s.

Looking at this and information presented below it is obvious that current technology is capable of detecting the Pioneer Anomaly.

The accuracy with which the gradient of a Doppler velocity against time plot can be calculated is dependent on the errors on the individual Doppler shift frequencies; each data point.  The error on the Doppler shift propagates to the Doppler residual and then onto the Doppler velocity.

The error on the Doppler measurement is given by a standard deviation on the Doppler velocity.  This section of the project characterises the magnitudes of some Doppler errors in X-band and then considers in the light of these how the accuracy of interpreting the observed data is constrained.

The signal received during an integration time is characteristic of an additive white Gaussian noise source (AWGN).  The AWGN is due to the fact that the Doppler velocity comes from a counting process and the result forms a Gaussian profile which has a statistically inherent distribution; Doppler jitter.  The distribution of the Gaussian is characterised by the standard deviation of the profile which can be calculated as a function of parameters associated with the telemetry link; equation 1.

(1)

The telemetry link parameters designed by this project which are appropriate to determine the Doppler jitter using equation 1 are shown in table 1.

 Designed Telemetry Link Parameters One sided noise equivalent loop bandwidth (BL) 10 Hz Uplink carrier power to noise spectral density ratio ([Pc/N0]UL) 27.4 dBHz 551 Hz Downlink carrier loop signal to noise ratio (ρL) 55.9 dBHz 390124.68 Hz Transponding ratio (G) 1.17 Integration time (T) 60 s Downlink carrier frequency X-band (fc) 8.45 GHz Doppler velocity error due to link jitter (σv) 1.05x10-5 ms-1 0.011 mms-1

Table 1, Telemetry link parameters used to calculate Doppler jitter (X-band).

Table 1 shows a Doppler jitter of 0.011 mm/s.  This can be compared to the quoted value of 0.03 mm/s for the NASA Deep Space Network[1].  The discrepancy is likely due to the simplified modelling of the noise power in the telemetry link.  Therefore in a more comprehensive analysis of the telemetry link the project value of the Doppler jitter is only going to increase towards the quoted value.

Along with Doppler jitter there is an error on the Doppler velocity measurement due to the instability of the generated reference frequencies.  The accuracy to which the reference frequency is generated is characterised by the Allan deviation of the oscillator (σy); equation 2.

(2)

The Allan deviation is a function of integration time, and is given for systems with respect to standard Doppler integration times.  Currently Allan deviations vary from 10-13 to 10-15.  The Small Deep Space Transponder (SDST) used as a baseline for the downlink model reports an Allan deviation of better than 10-13 over an integration time of 10s, and currently the Deep Space Network has an Allan deviation of 3.2x10-16 for an integration time of 1000s[2].  This design study assumes that the ESA ground stations have a similar Allan deviation.

The Allan deviation adds an error to the Doppler velocity, σv.

(3)vii

Equation 3 gives the error on the Doppler velocity over an observation time with number of samples N.  For the current design the error due to Allan deviation at the ground station is 0.0001 mm/s during one integration period of 1000s.

 Designed Allan Deviation Values Allan deviation (function of integration time) (σy) 3.20x10-16 Observation time (τ) 1000 s Integration time (T) 1000 s Number of samples (N) 1 Error on Doppler velocity due to Allan deviation (σv) 9.60x10-8 ms-1 0.0001 mms-1

Table 2, Parameters used in calculating the error on Doppler velocity due to Allan deviation.

Doppler jitter and Allan deviation are the two main inherent sources of noise considered.  Table 3 summarises the values of the errors on the measured Doppler velocity over an integration time of 1000s.

 System Error Values on Doppler Velocity (mm/s) Pioneer Anomaly Doppler Velocity 0.0009 Doppler jitter 0.0006 Allan Deviation 0.0001 Total Measurement 0.0009 ± 0.0006

Table 3, Summary of error values on Doppler Velocity and Pioneer Anomaly Doppler velocity over 1000s.

The error on a measured Doppler velocity due to Allan deviation is nine times smaller than the Pioneer Anomaly Doppler velocity and two thirds the size for Doppler jitter.  Therefore in one measurement the Pioneer Anomaly can potentially be observed but not with a high level of accuracy.  To achieve higher accuracy measurements are made over an observation time collecting a set of Doppler velocities.

Equation 1 shows that Doppler jitter is independent of observation time and equation 3 shows that the error due to Allan deviation is proportional to one over the square root of the observation time.  Therefore during the observation time the error on the Doppler velocity measurement decreases.

Graph 1, Plot showing manifestation of the Pioneer Anomaly in Doppler data (blue), along with the total error value as a function of observation time (red).

Graph 1 shows that below 1000 seconds the inherent Doppler errors mask the Pioneer Anomaly Doppler velocity but once the craft has been tracked for over 1000 seconds (one integration period) the Doppler error values are below that due to the Pioneer Anomaly.

[1] Thornton, C.L., Border J.S., 2003, Radiometric Tracking Techniques  for Deep Space Navigation, JPL Deep Space Communications and Navigation Series, Wiley

[2] Nieto, M.N., Turyshev, S.G., 2004, ‘Finding the Origin of the Pioneer Anomaly’, Class. Quantum Grav. 21, 4005-4023

Nieto, M.N., Turyshev, S.G., 2004, ‘Finding the Origin of the Pioneer Anomaly’, Class. Quantum Grav. 21, 4005-4023