6.4 Effects on the radio signal

It has been argued [301] that the cosmic expansion influences the measurement process via a change in the frequency of the traveling electromagnetic signals. However, one expects that taking all effects of the cosmic expansion on the frequency as well as on the Pioneer motion into account, the resulting acceleration is − vH0 and, thus, has the correct sign but is too small by a factor v∕c [177]. The ways in which the cosmic expansion might be responsible for aP vary considerably between the approaches. It is known [24Jump To The Next Citation Point27Jump To The Next Citation Point] that the very presence of the Pioneer anomalous acceleration contradicts the accurately known motion of the inner planets of our solar system. This motivated focusing on the effect of cosmic acceleration on the radio communication signal rather than on the spacecraft themselves. This mechanism might be able to overcome the apparent conflict that aP presents to modern solar system planetary ephemerides [24Jump To The Next Citation Point27Jump To The Next Citation Point].

6.4.1 Helicity-rotation coupling

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):

′ ω = ω ∓ Ω, (6.27 )
where ω is the signal frequency that would be observed in the absence of rotation, Ω is the rotational frequency, while ′ ω is the observed frequency. The sign is positive for a negative helicity wave, and vice versa, and the formula must be applied to each leg of the communication separately (e.g., to the uplink and downlink leg in the case of two-way or three-way Doppler data.)

For the Pioneer spacecraft, Ω ≃ 0.076 Hz (Pioneer 10) and Ω ≃ 0.13 Hz (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).

6.4.2 Clock acceleration

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 a P or a time acceleration a t that is connected with the former by the relationship aP = c at. This fact motivated Anderson et al. [27Jump To The Next Citation Point] 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 [27Jump To The Next Citation Point] 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 1 2 t → t + 2att. In particular, let any labeled time ta be given as

1- ( 2 2) ta − t0 → ta − t0 + 2at ta − t0 , (6.28 )
then the light time is
1 ( 2 2 ) Δt = treceived − tsent → Δt + -at treceived − tsent . (6.29 ) 2

Expression (6.29View Equation) mimics a line of sight acceleration of the spacecraft, and could be thought of as an expanding space model. Note that at 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 [27Jump To The Next Citation Point]. This fact motivated the discussion on the nature of the implied numerical relationship between the Hubble constant and aP (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 [27Jump To The Next Citation Point]; however, such an effect should also be observed in the radio signals from pulsars [198410], 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 [315316], 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 [199200].

6.4.3 Cosmological effects on radio signals

Lämmerzahl [176Jump To The Next Citation Point] 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 v, the cosmologically-induced acceleration aH would have the following form:

v- aH = vH = ccH. (6.30 )
Clearly, this acceleration is a factor of v∕c smaller that the observed Pioneer acceleration, for which aP ≃ cH. Therefore, Doppler tracking in an expanding universe cannot account for the observed Pioneer anomaly [73176].
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