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# Pioneer anomaly and Mercury's anomalous perihelion precession solved in a fluid quantum vacuum

Marco Fedi

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Marco Fedi. Pioneer anomaly and Mercury's anomalous perihelion precession solved in a fluid quantum vacuum. 2017. hal-01592480

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Distributed under a Creative Commons Attribution 4.0 International License
Pioneer anomaly
and Mercury’s anomalous perihelion precession
solved in a fluid quantum vacuum

Marco Fedi *

25 September 2017

**Abstract**

We assume that quantum vacuum is a bi-regime quantum fluid which remains superfluid under low shear stress and becomes dilatant as the shear stress increases towards a relativistic limit. We modify Stokes’s law for viscous friction resorting to Lorentz factor to express a divergent apparent viscosity. In this way we calculate an exact solution for the so-called Pioneer anomaly and we derive the formula used in general relativity to calculate the relativistic contribution to perihelia precessions. Clues for the existence of a bi-regime fluid quantum vacuum appear.

**Keywords**— quantum vacuum, dilatant vacuum, Pioneer anomaly, anomalous perihelia precessions, Stokes’s law, Lorentz factor

**PACS**— 95.55.n, 98.80.Qc, 03.75.Nt, 47.50.-d, 83.60.Fg, 03.30.+p, 04.20.-q

**Introduction: a bi-regime fluid quantum vacuum**

Along with several other authors, we consider quantum vacuum as a special, dark fluid [1, 2, 3, 4, 5, 6, 7, 8, 9], probably consisting of dark energy and dark matter (95% of the universe’s mass-energy) which undergo quantum hydrodynamic perturbation such as quantized vortices, but we introduce an important difference: we assume it is a bi-regime quantum fluid, which behaves as a superfluid under low shear stress (in the limit of classical physics) and becomes on the contrary dilatant for relativistic shear stress, i.e. as the velocity of the bodies which travel through it approaches that of light, according to Lorentz factor [10]. The reason for using Lorentz factor as the expression of vacuum’s diverging apparent viscosity is that it is impossible to surpass the speed of sound in dilatant fluids (unless their transient solid-like structure breaks) for bodies whose size is greater than that of the fluid’s molecular interstices. In quantum vacuum these interstices are likely at Planck or sub-Planck scale and this, along with the non-finite or cosmic-scale spatial extent of quantum vacuum, would make it impossible for any material body (even for a neutrino) to travel through it faster than light,

<footnote>*Ministero dell'Istruzione, dell'Università e della Ricerca (MIUR), Rome, Italy.
E-mail: marco.fedi.caruso@gmail.com</footnote>
<figure><img src="image_3.png" /><figcaption>Figure 1: Lorentz factor as the rheogram of quantum vacuum, by hypothesizing it passes from a superfluid to a dilatant regime under relativistic shear stress, when accelerated bodies approach the speed of sound through it. The asymptote represents the passage to a solid, impenetrable state, a situation in which only sound propagation is possible, which in fluid quantum vacuum corresponds to light as transverse phonon propagation in a quasi-lattice. Here *k* is an adimensional scale factor.</figcaption></figure>

as we verify in synchrotrons, where the divergence of a braking force acting in the opposite direction to the provided acceleration and due to a dilatant vacuum might be erroneously interpreted as the divergence of mass. This reasoning implies that light is the sound of a dilatant quantum vacuum (photon-phonon analogy in quantum vacuum) and indeed we have already considered this possibility as very likely [10]. In addition, revisiting the nature of light in a fluid vacuum seems able to solve critical issues in modern cosmology [21]. This would point at a quantum hydrodynamic foundation of Lorentz factor and of special relativity itself, which currently resorts to the insuperability of light purely as a matter of fact.

Within this framework, we modify Stokes's law for viscous friction to express dynamic viscosity not in a linear law but through the Lorentz factor, where the ratio translational velocity to speed of light indicates the behavior of apparent viscosity, whose increase becomes in this way nonlinear, as necessary for dilatant fluids, and asymptotic, as necessary for the case of quantum vacuum, as discussed above. In this view we interpret Lorentz factor as the rheogram of quantum vacuum (Fig. 1).

# 1 The Pioneer anomaly

Being the anomalous negative acceleration of the Pioneer spacecrafts 10 and 11 well-known (concrete investigations of the anomaly started in 1994 [12]), we do not summarize here this issue. However, before introducing our explanation from the point of view of a fluid quantum vacuum, we reflect that the commonly accepted explanation, i.e. an excessive deceleration due to the anisotropic recoil of thermal photons [11, 12, 13, 14, 15], is in our opinion neither reliable nor definitive yet, due to various approximations and assumptions. Even at the end of mission, the heat radiated
by the probe was in excess of 2 kW. Since the power needed to produce the reported acceleration is only ~65W, the anisotropy is only ~3% and a small error in computing the total radiated heat produces an unacceptable discrepancy in the recoil force [16]. Moreover, every revolution of the spinning spacecraft adds further discrepancies to measurements. Third, the largest amount of heat was radiated from the radioisotope thermoelectric generators, not from the louvers, being useless as regards the computation of the anomalous deceleration. Finally, without accurately knowing the dynamics of heat diffusion inside the probe [15], we cannot be sure that the anisotropic recoil of thermal photons from the louvers be the reason of the Pioneer anomaly. Scheffer [14] analyzes in detail four sources of anisotropic heat radiation and their possible contribution to the anomaly. The author assumes (a) a uniform internal temperature with closed louvers, while the modeling of the internal heat is considered difficult due to a lack of precise information and, on the other hand, other studies indicate the radiation from the louvers as a possible valid source of anisotropic contribution to the anomaly [17]; (b) the contribution from the not reflected photons coming from the radio antenna (at about 45 degree angle to the spin axis) is actually balanced by the symmetrical contribution in the direction of the Sun; (c) the radiation from the radioisotope heater units and from the radio-thermal generators are considered by the author difficult to calculate and are hypothesized; (d) the same uncertainties, assumptions and hypothetical modeling concern the antenna's solar reflectivity and are common features to all the studies done around the thermal hypothesis. In a last analysis we can therefore conclude that the thermal cause of the anomaly is possible by opting for suitable values in the calculations among many different scenarios of thermal diffusion but is not demonstrated yet, since an accurate thermal modeling is difficult [11, 14].

On the contrary, here we calculate the deceleration using Stokes's law for friction in a (Newtonian) viscous fluid

$$F_v = 6\pi r v \eta \quad (1)$$

where v is the translational velocity, r the radius of the object (the law considers a spherical object) and η is a coefficient of dynamic viscosity expressed in Pa·s. We calculate the Pioneer's anomalous deceleration as $a_P = -F_v/m$, where m is the mass of the spacecraft and the sign minus is due to the frictional (braking) force. Since we are considering a dilatant fluid, the relationship between velocity and viscosity is not linear but obeys, as assumed, Lorentz factor, dimensionally corrected by the unitary constant κ, as Lorentz factor represents here the apparent dynamic viscosity which nonlinearly increases with velocity

$$a_P = -\frac{F_v}{m} = -\frac{6\pi r \left( \frac{1}{\sqrt{1-\left(\frac{v}{c}\right)^2}} - 1 \right) \kappa}{m} = -\frac{6\pi \cdot 1.37 \text{ m} \cdot \left( \frac{1}{\sqrt{1-\left(\frac{36657 \text{ m} \cdot \text{s}^{-1}}{299792458 \text{ m} \cdot \text{s}^{-1}}\right)^2}} - 1 \right) \cdot 1 \text{ Kg} \cdot \text{s}^{-2}}{222 \text{ Kg}} \quad (2) \\ = -8.7 \times 10^{-10} \text{ m} \cdot \text{s}^{-2}$$

In the calculations we consider: the mass of the spacecraft (258 kg) minus the mass of the burned fuel (36 kg), the radius of the antenna and the maximum speed of the probe after the swing-by caused by Jupiter. As we see the result is in agreement with
the measured value expressing the anomalous negative acceleration

$$a_P = -(8.74 \pm 1.33) \times 10^{-10} \, \text{m} \cdot \text{s}^{-2} \qquad (3)$$

## 2 Mercury's anomalous perihelion precession

Net of classical gravitational contributions, perihelia precessions show an anomalous positive contribution, which is particularly evident for the planet Mercury. In general relativity this anomalous (relativistic) contribution to the precession is given by a formula [18] which can be observed in three equivalent forms

$$\Delta\phi = \frac{24\pi^3 a^2}{T^2(1-e^2)c^2} = 6\pi \left(\frac{v}{c}\right)^2 \frac{1}{1-e^2} = 6\pi \frac{GM}{a(1-e^2)c^2} \qquad (4)$$

where $\Delta\phi$ expresses the relativistic contribution to perihelia precessions in radians per revolution corresponding, using the data of Mercury, to the known value of ~43" per century (or $5.018 \times 10^{-7}\text{rad/rev}$), a is the semi-major axis, T the orbital period and $e = 0.205$ the orbital eccentricity. In the expression in the center, we have replaced the expression for orbital velocity $v = 2\pi a/T$ and in that on the left we have used the second cosmic velocity $v = \sqrt{GM/a}$, where $GM/a$ is the classical gravitational potential, also putting $a = r = p$ [19], where p is the semi-latus rectum (elliptic parameter).

We hypothesize that the anomaly be due also in this case to the interaction with a fluid quantum vacuum and again we start from Stokes's formula (1). Since we treat Mercury as a point mass, we do not consider here the direct proportionality to the radius. To express the relationship velocity/viscosity we resort as above to Lorentz factor in the form $\gamma - 1$ and the expression becomes

$$6\pi \left( \frac{1}{\sqrt{1 - \frac{GM}{ac^2}}} - 1 \right) = 6\pi (\gamma - 1) \qquad (5)$$

Taking into account [19] we believe that the by the authors derived result (1/2 that of general relativity) is due to the fact that the relativistic contribution from special relativity occurs twice a orbital revolution but this fact remains hidden by resorting to the mean orbital velocity in the formula, which levels the effect along the whole orbit: actually, in the special relativity context, along the semi-orbit from aphelion to perihelion the positive acceleration of the planet causes a relativistic mass increase and a consequent slight increase of the centripetal acceleration due to the increased gravitational attraction. While in the semi-orbit from perihelion to aphelion, the orbital velocity decreases, along with the relativistic mass and the gravitational attraction, letting the planet move further in the direction of the precession. For this reason we multiply the effect by two and still from [19], using Taylor we resort to the approximation

$$2(\gamma - 1) \approx \left(\frac{v}{c}\right)^2 \qquad (6)$$

thus the expression reads

$$6\pi \left(\frac{v}{c}\right)^2 = 6\pi \frac{GM}{ac^2} \qquad (7)$$
Finally, since we consider motion in an elliptic orbit, we have to use the elliptic pa- rameter, correcting a into a (1 − e $^{2}$) and we obtain the expression of general relativity (4)

$$
6\pi \frac{GM}{a(1-e^2)c^2} \qquad (8)
$$

Conclusion

We show that an approach to quantum vacuum which considers it as a bi-regime quan-
tum fluid becoming dilatant under relativistic shear stress, may solve the known cases
of gravitational anomalies, unexplained by classical physics. Here exact solutions for
both the Pioneer anomaly and the perihelia precessions are derived. We also understand
that such an approach to quantum vacuum is claiming to become the quantum hydrody-
namic foundation of special and general relativity, where a geometric spacetime would
give way to the hydrodynamics of a fluid quantum vacuum. Further investigations and
studies of other cases are therefore suggested.

Acknowledgements

The author thanks Valery Sbitnev for interesting discussions concerning the Pioneer anomaly and Ali Taani for having provided useful references about Mercury’s anomalous perihelion precession in special relativity.

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