Anomalous Acceleration of Pioneer 10 and 11

Signal Propagation Effects

In the last 10 years or so, the tracking data for the Pioneer
10 and 11 space probes on their way out of the solar system appear to reveal an
unexpected slowdown of their velocity i.e. an anomalous acceleration towards the
sun which can apparently not be explained by known forces (see the paper by Anderson et al.
(note: the PDF version of the paper has a size of 1.4MB)). The figure below
(which is taken from this paper) gives a plot of the observed unexplained part
of the acceleration against distance from the sun. It shows that the
acceleration is constant (or nearly constant) over the whole range where the
error bars are small enough so that the data can be considered as significant
(i.e. between about 20 and 50 AU).

These data are derived by comparing the Doppler shift of the radar
tracking signal beamed from the earth to the space probe and it is obvious that
any incorrect assumptions as to the exact distance of the spacecraft could
distort the derived values. The point is that, as shown on my page regarding the
Speed of Light, the
usual assumption for the travel time of a light signal between source and
receiver would lead to an underestimation of the distance if the two are moving
relatively to each other. In this case, the mismatch at distance s(t) would be

(1)     Δs=-s(t).v(t)/c     ,

where v(t) is the spacecraft velocity.
The apparent acceleration is
therefore

(2)     Δa(t)=d2(Δs)/dt2 = -
1/c.d2[s(t).v(t)]/dt2    .

If one evaluates the differential one gets

(3)     Δa(t)=-a(t).3.v(t)/c -
s(t)/c.da(t)/dt    ,
where a(t)=dv(t)/dt is the
original acceleration due to whatever forces are acting on the probe. If one
assumes a(t) to be given just by the gravitational field of the sun's mass M
,i.e.

(4)     a(t)=GM/s2(t)    ,
and
assumes furthermore that v is roughly constant over the considered range, i.e.

(5)     s(t)=v.t     ,
one finds
that the second term in Eq.(3)

(6)     s(t)/c.da(t)/dt =
-2.a(t).v/c    ,
and therefore from (3)

(7)     Δa(t)=-a(t).v/c    .

Assuming that a(t) is given by the sun's gravitational acceleration at a
distance of 20 AU and that v=10 km/sec one gets

Δa(20AU) = -4.9.10-8
cm/sec2 ,
and at 50 AU

Δa(50AU) = -7.9.10-9
cm/sec2
The apparent acceleration has therefore the observed
order of magnitude but decreases with distance and is directed away from the
sun. If one subtracts this acceleration from the constant observed acceleration
of +8.10-8 cm/sec2, one obtains

Δa(20AU)(data) = 1.3.10-7
cm/sec2    ,
and at 50 AU

Δa(50AU)(data) = 8.8.10-8
cm/sec2    .
Corrected for the error due to the
signal propagation, the anomalous acceleration is now not constant anymore (as
displayed in Andersons et al.'s analysis) but decreases with distance. This
should substantially alter the situation regarding which effects might be
responsible. However, I would also like to mention that Anderson et al. quote
the statistical error of the measurements as 2.10-8
cm/sec2 . This is actually not much smaller than the alleged
anomalous acceleration, so one has to ask if there has probably a constant
factor been neglected in the analysis, i.e. if the whole effect is probably not
some error noise, probably caused as an artefact of the numerical data analysis
similar to the flaws in the WMAP Data Analysis. The
apparent decrease of the acceleration which follows from the above analysis
could then be probably be accidentally offset by an incorrect modelling of the
radiation pressure force.

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