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This principle is described here following the work of Claus Lämmerzahl and his collaborators (Lämmerzahl et al, 2008). If a radio signal with the frequency ν₀ is sent from the Earth to the spacecraft, this last one will receive the signal with a different frequency, due to the so-called Doppler Effect

(1)

where v is the magnitude of the spacecraft relative velocity and c is the speed of light. Now, the spacecraft antenna will reflect back this signal, which will be received on Earth with another frequency, due to the very same Doppler Effect. This frequency will be expressed by the same formula as above, so that the receiver on Earth has the signal back with a frequency

(2)

Thus, by measuring the difference between the received and the sent signal one has information on the speed of the spacecraft. From the comparison of the times of sending and receiving one has also information upon the position of the craft. This was, in broad lines, the principle of tracking and locating the Pioneers. There are, of course, details which make extraction of the information from the raw data hard, but those are well detailed in dedicated works and are not of concern here.

The Pioneer anomaly appears in the following way: the difference of the two signal frequencies – the one received and the one sent – is given by

(3)

In the range of the spacecraft relative speeds, which is between 10 and 40 km/s, the ratio β is maximum 1.5×10^-4, so that the relative frequency deviation is well represented by the approximation

(4)

This gives a value of the spacecraft velocity

(5)

This “measured” velocity – in fact the ratio β – can be compared with that calculated according to our current theoretical knowledge. When this is done for both Pioneers, a constant bias appears over time, with the slope

(6)

This is translated in a constant variation of the velocity (5) of the spacecraft, amounting to the value of

(7)

pointing in the general direction of the inner solar system. This is the Pioneer anomaly.

Reference

C. Lämmerzahl, O. Preuss, H. Dittus (2008): Is the Physics within the Solar System Really Understood?, arxiv: 0604.052 v1