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 {\it et al} in 1998. What they observed is similar to the effect that would have either (i) an anomalous acceleration $a_{\rm P}$ of the ship towards the Sun or (ii) an acceleration of the clocks $a_{\rm t}=a_{\rm P}/c$. 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 $2a_{\rm t}={\rm d}\Psi /{\rm d}t ={\rm d}^2\tau_{\rm clocks} /{\rm d}t^2$, evaluated at present time $t_0$, where $t$ and $\tau_{\rm clocks}$ are the coordinate time and the time measured by the clocks, respectively. The result of a simple estimate gives the value $a_{\rm t}\simeq 1.8\times 10^{-18}{s}^{-1}$, while Anderson {\it et al} suggested $a_{\rm t}= (2.9\pm 0.4)\times 10^{-18}{s}^{-1}$ on the basis of their observations. The calculation are performed near the Newtonian limit but in the frame of general relativity.