The radio signal used for communicating with Pioneer 10 and 11 (and indeed, used routinely with other spacecraft) is circularly polarized. Therefore, the question naturally arises as to whether the coupling between the helicity of the radio signal and the rotation of either the transmitter or the receiver could contribute to the observed Doppler anomaly in the Pioneer radio signal [196]. To first order, this coupling can increase or decrease the frequency of a radio signal by the rotational frequency of the transmitter (or receiver):
where For the Pioneer spacecraft, (Pioneer 10) and
(Pioneer 11). However,
Anderson and Mashhoon [30] note that this effect cannot account for the Pioneer anomaly. The effect of
rotation on the radio signal is already phenomenologically incorporated into the Doppler data analysis (see
Section 4.5.4).
The fact that the anomaly was discovered using Doppler techniques leaves duality in the nature of the
detected signal – it is either true physical acceleration or a time acceleration
that is
connected with the former by the relationship
. This fact motivated Anderson et
al. [27
] to try to look for purely phenomenological “time” distortions that might fit the Pioneer
data. The question was “is there any evidence that some kind of ‘time acceleration’ is being
seen?”.
A number of models were investigated and discarded for various reasons (see [27] for discussion), but
there was one model that was especially interesting. This model adds a term that is quadratic in time to the
light time, as seen by the DSN station as
. In particular, let any labeled time
be given
as
Expression (6.29) mimics a line of sight acceleration of the spacecraft, and could be thought of as an
expanding space model. Note that
affects only the data, not the trajectory. It was pointed out by
Anderson et al. that this model fits both Doppler and range very well for several spacecraft used in
their study [27
]. This fact motivated the discussion on the nature of the implied numerical
relationship between the Hubble constant and
(Section 6.6.2). To investigate further the
nature of this relation one would need to check the data of other spacecraft, compare modern
clocks with accuracy much higher then that used in the navigation of the Pioneers, as well
as the data on millisecond binary pulsars. Presently, not all of these venues are yet properly
explored.
Rañada [302] investigated the effect of a background gravitational potential that pervades the universe
and is increasing because of the expansion, provoking a drift of clocks [27]; however, such an effect should
also be observed in the radio signals from pulsars [198, 410], which is not the case. Further refining their
argument, Rañada and and Tiemblo [303] investigated the nonequivalence of atomic and astronomical
times and concluded that these times could be accelerating adiabatically with respect to one
another.
Ostvang [276] proposes that cosmic expansion applies directly to gravitationally bound systems according to the quasi-metric framework. According to [315, 316], the scale factor of the spacetime background would cause an anomaly in the frequency. The cosmological constant has also been invoked to produce acceleration [268] or a gravitational frequency shift [199, 200].
Lämmerzahl [176] considered the possibility that an expanding universe may have an effect on the
Doppler microwave signals traveling in the solar system. The basic question is whether or not, if it exists,
the coupling of the expansion of the universe to light has an observable effect. It was shown that for a
spacecraft moving with velocity
, the cosmologically-induced acceleration
would have the following
form:
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