The pioneer anomaly as acceleration of the clocks

This work proposes an explanation of the Pioneer anomaly, the unmodelled and as yet unexplained blueshift detected in the microwave signal of the Pioneer 10 and other spaceships by Anderson et al. in 1998. What they observed is similar to the effect that would have either ( i) an anomalous acceleration a(P) the ship towards the Sun, or ( ii) an acceleration of the clocks a(t) = a(P) /c. The second alternative is investigated here, with a phenomenological model in which the anomaly is an effect of the background gravitational potential Psi(t) that pervades all the universe and is increasing because of the expansion. It is shown that 2a(t) = d Psi/ dt = d(2)tau(clocks)/dt(2), evaluated at present time t(0), where t and tau(clocks) are the coordinate time and the time measured by the atomic clocks, respectively. The result of a simple estimate gives the value a(t) similar or equal to 1.8 x 10(-18) s(-1), while Anderson et al. suggested a(t) = (2.9 +/- 0.4) x 10(-18) s(-1) on the basis of their observations. The calculation are performed near the Newtonian limit but in the frame of general relativity.