Detection of the Pioneer Anomaly in Doppler Data
The Pioneer Anomaly was inferred from an un-modelled Doppler shift in signals received from both Pioneer 10 & 11. A Doppler shift is where the frequency of a radio signal is changed by the velocity of the transmitter relative to the receiver. Similar to changes in sound as a train passes through a train station. So because the Pioneer craft are moving away from the Earth radio signals from them have their frequency decreased.
But over time the Doppler shift slowly varied from that predicted by the velocity of the Pioneer craft. A possible explanation for this is that an acceleration is affecting the velocity of the Pioneer craft. The work carried out to infer this anomalous acceleration (Pioneer Anomaly) is presented in detail below.
To summarise: although the anomaly directly affects the craft’s velocity to analyse it a value known as Doppler velocity is used. The Doppler velocity for Pioneer 10 over 60 seconds is 5.25x10-5 mm/s. This can be directly inferred as an acceleration of 8.168x10-10 ms-2 in a direction towards the sun.
Doppler tracking is where a radio signal is transmitted from the Earth to a spacecraft. The signal is then coherently transponded and sent back to the Earth.
The received frequency (fR) is a function of the transmitted frequency (fT) and the velocity of the craft (range-rate, v), shown in equation 1.
The frequency change of the received signal is compared to the transmitted signal over a period known as the integration time (T). The change in frequency is called the Doppler shift which is directly related to the line of sight velocity of the craft.
A Doppler frequency is recorded at the end of each integration period. The craft is tracked for an observation period (τ), during which time the Doppler frequency measurements are made. At the end of an observation period there is a set of Doppler frequency shift measurements, numbering τ/T.
The received frequency (fobs) is compared to a modelled frequency (fmod), to give a Doppler residual (Δf).
If the Doppler residual is zero then there are no un-modelled accelerations acting on the craft. If the residual is not zero there is some unknown acceleration acting on the craft. In the case of the Pioneer Anomaly between 1987 and 1998 the Pioneer Doppler residual varied at a rate of 5.99x10-9 Hzs-1, this is known as a Doppler drift. A constant Doppler drift can be interpreted as a constant acceleration, ap.
The Pioneer craft Doppler drift gives an un-modelled acceleration of -8.168x10-10 ms-2. This un-modelled acceleration is known as the Pioneer Anomaly. This value varies from that quoted in the literature because this value is from an incomplete model. The value quoted by Anderson et al[i]. takes into account other forces acting on the Pioneer craft, radio reaction force, mass uncertainty.
It is the Doppler residual converted into a Doppler velocity which is used for analysis of the anomalous acceleration. For Pioneer 10 the Doppler residual is 7.7x10-7 Hz for an integration time of 60s. The residual is converted into a velocity by a factor of 1 Hz = -68.2 mms-1.
The measured Doppler velocity can be modelled between 1987 and 1998.
Graph 1, Plot of the Doppler velocity of Pioneer 10 between 1987 and 1998, according to the model presented here.
This plot corresponds with that presented by Anderson et al.[ii] in 2002 in a report of the Pioneer Anomaly.
Graph 1 shows that over an extended observation period the Doppler velocity increases to relatively sizeable values. Over the 11.5 years of this initial observation Pioneer 10 became 57 500 Km out of position compared to a distance travelled of 30 A.U.
[i] Anderson, J.D., et al., 2002, ‘Study of the anomalous acceleration of Pioneer 10 and 11’, Phys. Rev. D 65, 082004
‘Indication from Pioneer 10/11, Galileo, and Ulysses Data, of an Apparent
Anomalous, Weak, Long-Range Acceleration’, Phys. Rev. Lett. 81, 2858