We review the fundamental results of a new cosmological model, based on conformal gravity,
and apply them to the analysis of the early data of the Pioneer anomaly.
We show that our conformal cosmology can naturally explain the anomalous acceleration of the
Pioneer 10 and 11 spacecrafts, in terms of a local blueshift region extending around the solar system
and therefore affecting the frequencies of the navigational radio signals exchanged between Earth
and the spacecraft. By using our model, we explain the numerical coincidence between the value of the anomalous
acceleration and the Hubble constant at the present epoch and also confirm our previous determination of the cosmological parameters

The Pioneer 10 and 11 spacecrafts were launched in the early 1970s, to conduct explorations in the region of the solar system beyond the orbit of Mars and to perform close observations of Jupiter. They were also the first spacecraft to explore the outer solar system and to send back to Earth their navigational signals for almost thirty years (for a review see [

In recent years, the orbits of Pioneer 10 and 11 were reconstructed very accurately, by using the original radiometric Doppler tracking data, based on the signals exchanged between the spacecraft and NASA’s terrestrial tracking stations. This reconstruction yielded a persistent discrepancy between the observed and predicted data, equivalent to an unexplained small acceleration of the spacecraft in the direction of the Sun. This effect is evidenced by measuring a small frequency shift (toward higher frequencies, i.e., a “blueshift”) of the signal reaching us from the spacecraft. The nature of this anomalous acceleration or of the related blueshift remains unexplained; this effect has become known as the “Pioneer anomaly” ([

This is not the only known gravitational anomaly in the solar system, since several others are currently under investigation (for general reviews see [

The importance of all these effects is not related to how they affect the spacecraft navigation, since they all produce very small corrections to the orbits, but to the possibility that these anomalies might be an indication of new gravitational physics. In particular, several nonconventional explanations of these effects have been proposed (see general discussion in [

In this line of reasoning, alternative gravitational theories such as conformal gravity (CG), originally proposed by Weyl in 1918 ([

Following the original CG, we have recently studied an alternative approach to these models which was named “kinematical conformal cosmology” [

A preliminary analysis also performed in our second paper [

In the next section, we will briefly review our CC solutions, showing how a local blueshift region can naturally emerge, while, in Section

In our first CC paper [

We then considered regions far away from matter distributions, thus ignoring the matter dependent

It was precisely this connection between the two solutions which prompted us to consider the CG static, spherically symmetric solution as an alternative description of the standard cosmological evolution, based on the RW metric. In other words, the CG static solution might also contain information about the cosmological redshift, the expansion of the Universe, and so forth, and constitute an alternative approach to cosmology. In particular, the CG expressions in (

Therefore, we postulated in [

In the previous equation, the redshift factor

In Figure

Our preferred solution in Figure

Before we proceed to analyze this possible explanation for the anomaly, we recall a few more results obtained in our second paper [

The signs of the quantities in (

Therefore, for each value of

In particular, following (

We conclude this section by noting that the values of our parameters (

In the previous section, we briefly reviewed our conformal cosmology and outlined the reasons why we consider the

This could be a serious problem for our model, since we do not observe blueshift of nearby astrophysical objects except for the one caused by the peculiar velocities of nearby galaxies, presumably due to standard Doppler shift. However, as already mentioned in Section

This is a small frequency drift (blueshift), observed analyzing the navigational data of the Pioneer 10-11 spacecraft, received from distances between 20 and 70 AU (astronomical units) from the Sun, while these spacecraft were exploring the outer solar system. This anomaly is usually reported as a positive rate of change of the signal frequency,

An attempt was made to detect such anomaly also in the radiometric data from other spacecraft traveling at the outskirts of the solar system, such as the Galileo and Ulysses missions [^{2} was seen in the data. Other spacecraft, such as the New Horizons mission to Pluto, launched in 2006, might provide new data in the near future. These discoveries prompted a complete reanalysis of all the historical navigational data of these space missions, which is currently underway ([

Currently, the origin and nature of this anomaly remains unexplained; all possible sources of systematic errors have been considered ([

Several other papers ([

Although the anomaly can be caused by these standard physical effects, we will try in the following to explain its origin by using the cosmological model outlined in the previous section. The phenomenology of the Pioneer anomaly is related to a complex exchange of radiometric signals between the tracking stations on Earth (of the deep space network (DSN)) and the spacecraft, using S-band Doppler frequencies (1.55–5.20 GHz). Typically, an uplink signal is sent from the DSN to the spacecraft at a frequency of 2.11 GHz, based on a very stable hydrogen maser system, then an S-band transponder on board the spacecraft applies an exact and fixed turn-around ratio of 240/221 to the uplink signal, so that the Pioneer returns a downlink signal at a slightly different frequency of about 2.29 GHz, to avoid interference with the uplink one.

This procedure is known as a two-way Doppler coherent mode and allows for very precise tracking of the spacecraft, since the returning signal is directly compared to the original one. On the contrary, a one-way Doppler signal (with a fixed signal source on the spacecraft, whose frequency cannot be monitored for accuracy) is less effective. This type of tracking system added to the propulsion and navigational characteristics of the Pioneer spaceship (especially the presence of a spin-stabilization system) resulted in a very good acceleration sensitivity of about 10^{−8} cm/s^{2}, once the influence of solar radiation pressure can be neglected (for distances

The DSN station acquires the downlink signal after a time delay ranging from a few minutes to some hours, depending on the distance involved, and compares it to the reference frequency to determine the Doppler shift due to the actual motion of the spacecraft. The navigational software can also model with great precision the expected frequency of the signal returned from the Pioneer, which should coincide with the one observed on Earth. As already mentioned, a discrepancy was found, corresponding to the values in (

The Pioneer anomaly was first reported ([

In the standard analysis of the Pioneer anomaly, the signal coming back to Earth is affected by the relativistic Doppler effect. Following this model,

Since we have a two-way system, the Doppler shift involved is actually double, so we can use the previous equation but with

On the contrary, a different frequency is observed,

The anomalous acceleration

These frequency differences

This result immediately explains the often cited “numerical coincidence,” that is, the simple relation

The value of

Following (

In Figure

Early data for Pioneer 10/11 acceleration as a function of heliocentric distance. The average value of the anomalous acceleration is indicated in red-dashed, together with its error range (green-dotted). We also show linear fits of the data, which allow for the determination of our cosmological parameters

These two sets of data clearly show a possible decrease of the Pioneer anomaly (in the absolute value

If our conformal cosmology is the origin of the Pioneer anomaly, and not the thermal recoil force mentioned at the beginning of this section, our “jerk” equation (

We computed the slopes of our two linear fits in Figure

Another type of analysis is illustrated in Figure

Early data for Pioneer 10/11 acceleration as a function of heliocentric distance. The average value of the anomalous acceleration is indicated in red-dashed, together with its error range (green-dotted). We also show full conformal cosmology fits of the data, which allow for a better determination of our cosmological parameters

Again, in Figure

Comparing the results in the last two equations with those for

As already mentioned at the beginning of this paper, a new analysis of extended Pioneer data has recently appeared [

In the previous sections, we discussed how conformal cosmology provides a natural explanation for the Pioneer anomalous acceleration, in both magnitude and direction (i.e., the negative sign of the radial acceleration). We also explained the “numerical coincidence,” connecting

We first remark that a new analysis of rotational velocity data for spiral galaxies, based on conformal gravity, has recently appeared ([

The values of the dimensionful parameters ^{−1} and ^{−2}). This difference could be due, as we explained in [

However, it is instructive to compute the dimensionless

Conformal gravity considers local gravitational effects as being due to the local potential ^{−20}, while, at a distance of 100 AU (outer solar system), the same ratio is ~10^{−16}. The galactic potential, related to another conformal gravity term of the form

In particular, more recent studies have focused their attention on the critical issue of the influence that a gravitational Pioneer anomaly acceleration would have on the motion of bodies in the solar system, such as inner and outer planets, comets, and asteroids ([

To further clarify the issue, in our analysis of the Pioneer anomaly, we used the reported values of the anomalous acceleration

In this way, we also overcome the original objection, reported in [

In our view, precision ranging measurements with radio signals or lasers, based on the round-trip travel time from Earth to other bodies in the solar system, would not show any anomalous effect because the speed of light is not affected by our cosmological model and the corrections to the dynamics of the solar system due to conformal gravity are negligible.

On the contrary, we would observe an effect similar to the anomalous acceleration for a spacecraft, a planet, or any other object in the solar system, if we were to study its motion through Doppler frequency ranging, because of the intrinsic differences in frequency or wavelength for light emitted at different spacetime positions, due to our cosmological model.

A similar discussion can be done regarding possible explanations of the Pioneer anomaly of cosmological origin. Recent studies based on the standard Friedmann-Lemaitre metric ([

The size of the local blueshift region, which in our model is responsible for the frequency differences, can be easily estimated by using (

The maximum blueshift effect would be seen at

We also want to compare our estimates of the rate of change of the anomalous acceleration (i.e., the jerk

The second independent study was done by Toth [

Finally, we wish to comment briefly on the recent analysis of extended Pioneer data by Turyshev and collaborators [

Pioneer 10: ^{2},

In conclusion, the detailed analysis of the Pioneer anomaly presented in this work has indicated that our conformal cosmology might be the origin of this effect, while conformal gravity alone cannot account for the anomalous acceleration of the spacecraft. If our analysis is correct, it explains naturally the numerical coincidence between the Pioneer acceleration and the Hubble constant, including the signs of these quantities. In addition, we confirm our previous evaluations of the cosmological parameters,

This work was supported by a grant from the Frank R. Seaver College of Science and Engineering, Loyola Marymount University. The author would like to acknowledge Dr. S. Turyshev and Dr. P. Mannheim for very useful discussions and advice on the subject. The author also thanks the anonymous reviewers for the useful comments which helped improve the final version of the paper.