The tracking data of some interplanetary space craft (in particular Pioneer 10 and 11) have revealed an unexplained acceleration anomaly (see Anderson et al.). On my page Explanation of the Pioneer 10 and 11 Acceleration Anomaly, I have suggested an explanation for this in terms of the earth's rotation, but there is a further issue which needs to be taken into account in this context, which is the correct interpretation of the invariance of the speed of light with regard to the communication with the space craft:
the Pioneer velocity and acceleration data are derived by observing the Doppler phase shift of the radar tracking signal beamed from the earth to the space probe and back. Since the Doppler shift is directly proportional to the velocity, its time derivative is an indicator of the acceleration. This can be compared to the theoretical acceleration expected at the presumed location of the space probe. Any difference between the two would suggest that additional forces are acting on the probe which have not been taken into account in the orbit calculations. However, this conclusion rests on the assumption that the 'light distance' (which is what is relevant for the Doppler phase shift) is actually identical to the distance of the space craft, which would only be the case if the latter would be at rest relatively to the earth. As it is moving however, the two are not identical: a signal sent out from earth when the probe is at a distance s is received after time T=s/c regardless of the velocity of the space craft because of the invariance of the speed of light (see the page regarding the Speed of Light for more). Now during the time T, the probe will travel a further distance v^{.}T (assuming it has a constant velocity v within the time interval T), i.e. it will actually be at a geometrical distance s^{.}(1+v/c) when the signal is received. Thus the 'light distance' (which is effectively what is being measured) is smaller than the actual distance of the space probe (which is what the orbit calculations are referring to) by an an amount

The apparent acceleration is therefore

(1) Δs=-s(t)^{.}v(t)/c ,

The apparent acceleration is therefore

(2) Δa(t)=d^{2}(Δs)/dt^{2} = - 1/c^{.}d^{2}[s(t)^{.}v(t)]/dt^{2} .

(3) Δa(t)=-a(t)^{.}3^{.}v(t)/c - s(t)/c^{.}da(t)/dt ,

(4) a(t)=GM/s^{2}(t) ,

(5) s(t)=v^{.}t ,

(6) s(t)/c^{.}da(t)/dt = -2^{.}a(t)^{.}v/c ,

(7) **Δa(t)=-a(t) ^{.}v/c ** .

**Δa(20AU) = -4.9 ^{.}10^{-8} cm/sec^{2} **,

**Δa(50AU) = -7.9 ^{.}10^{-9} cm/sec^{2} **.

Thomas Smid (M.Sc. Physics, Ph.D. Astronomy)

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See also my sister site https://www.plasmaphysics.org.uk