The Flyby Anomaly and the Effect of a Topological Torsion Current Mario J. Pinheiro1 aDepartment of Physics, Instituto Superior T´ecnico - IST, Universidade de Lisboa - UL, Av. Rovisco Pais, 1049-001 Lisboa, Portugal Abstract A new variational technique determines the general condition of equilibrium of a rotating gravitational or electromagnetic system (or both) and provides a modified dynamical equation of motion from where it emerges a so-far un- forseen topological torsion current (TTC) [Mario J. Pinheiro (2013) 'A Varia- tional Method in Out-of-Equilibrium Physical Systems', Scientific Reports 3, Article number: 3454]. We suggest that the TTC may explain, in a simple and direct way, the anomalous acceleration detected in spacecrafts during close planetary flybys. In addition, we theorize that TTC may represent a novel rela- tionship between linear momentum and angular motion through the agency of a vector potential. Keywords: Variational Methods in Classical Mechanics; Statistical physics, thermodynamics, and nonlinear dynamical systems; Celestial mechanics (including n-body problems); Relativity and gravitation 1. Introduction Flyby (or swing-by, gravitational slingshot, or gravity assist maneuver) is a well-known method in interplanetary spaceflight to alter the path and the speed of a spacecraft using the gravity of a planet or other astronomical object IRevised manuscript URL: http://mjpinheiro.weebly.com/ (Mario J. Pinheiro) [email protected] Preprint submitted to Physics Letters a March 20, 2016 5 (see, e.g., Ref. [1]). The rescue of the Apollo crew in 1970 was the first flyby maneuver ever did, using the Lunar flyby [2]. But the flyby anomaly is one among other, possibly related, several astro- metric anomalies that are referred in the technical literature, such as the change of the solar mass over time M_ (changes that result from a balance between the 10 mass loss due to radiation and solar wind compensated by falling materials con- tained in comets, rocks and asteroids) leading to the observation of a decrease of −14 the heliocentric gravitation constant per year GM_ =GM = (−5:0 ± 4:1):10 per year and a variation of the astronomical unit by approximately 10 m per century [3]. Quite surprisingly, dark matter does not have a gravitational in- 15 fluence in the solar system because its density is very low [4]. The angular 41 momentum of the Sun seems to be smaller than expected (S ≤ 0:95 × 10 kg m2 s−1) unless the Sun's gravitomagnetic force is included [5, 6]. The anoma- lous behavior of the Saturnian perihelion cannot be explained in the framework of the standard Newtonian and Einsteinian General Theory of Relativity [7], 20 also suggesting the need of new physics or the effect of an external tidal poten- tial acting on the Solar System possibly due to a new hypothetical huge body, Tyche [8, 9]. The phenomenological modification of Newtonian dynamics pro- posed by MOND doesn't offer a satisfying explanation for Cassini spacecraft anomaly [8]. The Faint Young Sun Paradox [10, 11, 12] can possibly be accom- 25 modated within a certain general class of gravitational theories with nonminimal coupling between metric and matter predicting a secular variation of the Earth heliocentric distance [13, 14]. Recent analysis of a Lunar Laser Ranging data record revealed an anomalous increase of the eccentricity rate of the lunar orbit [15, 16, 14, 17]. This effect 30 is not related to a possible change of the speed of light [16] or some dissipation at the lunar core and mantle [15], but possibly non-tidal explanations can be viable [17]. Astrometric data points to the existence of at least four unexplained anoma- lies, from the small and constant Doppler frequency drift shown by the radio- 35 metric data from Pioneer 10 and 11, which can be interpreted as a uniform 2 −8 2 acceleration of aP = (8:74±1:33)10 cm=s towards the Sun found in the data of both spacecraft when they were at a distance of 20 au from the Sun [18, 19, 20, 21, 22, 23] to the disturbing observation that a number of satellites in Earth flyby have undergone mysterious energy changes [22]. This effect is essentially a 40 slight departure from Newtonian acceleration (see also Ref. [24] for an overview of unexplained phenomena within our Solar System and in the universe). As already shown in several ways [18, 20], the effect is not a real gravitational phenomenon that certainly would have affected other major bodies of the Solar System. The possibility that this uniform Sunward acceleration, such as the one 45 experienced by the Pioneer spacecraft, might have a gravitational nature was shown to be erroneous [25, 26, 27, 28], and could not even affect the motion of the outer planets of the Solar System [29, 30]. Rindler-type extra-acceleration on test particles was ruled out altogether because it would affect the main features of the Oort cloud [31]. Furthermore, exotic physics is probably not affecting 50 Pioneer spacecraft trajectories [32]. Also proposed was the possibility the Pioneer anomaly might have a different non-gravitational origin, such as a recoil force associated with an anisotropic emission of thermal radiation off the spacecraft [33, 34, 35, 36, 37, 38, 39]. As discussed earlier, a secular change in the astronomical unit (au) d(au)dt = −1 55 7 ± 2 m cy [40] was reported [41, 40, 42] and several explanations were pro- posed, among them, the change in the moment of inertia of the Sun due to radiative mass loss [43] but the possible variation of the dark matter density was ruled out [44]. The huge importance of the problem and the uncertainties related to the causes of its variation lead to the proposal of fixing the value of 60 au [45, 46, 47]. Other proposed explanations for this effect include an adiabatic acceleration of light due to an adiabatic decreasing of the permeability and per- mittivity of empty space [48]; the dilaton-like Jordan-Brans-Dicke scalar field as the source of dark energy, which introduces a new term of force with mag- 2 nitude aP = Fr=m = −c =RH (RH is the Hubble scale), (see Ref. [49]); light 65 speed anisotropy [50] based on Lorentz space-time interpretation and resorting from the earlier measurement of D. C. Miller (see also Ref. [51] which gives an 3 interesting reformulation of the special theory of relativity); and a computer modeling technique called the Phong reflection model [52], which explains the effect as due to the heat reflected from the main compartment, though this 70 explanation still needs confirmation. The flyby anomaly appears as a shift in the Doppler data of Earth-flybys of several spacecrafts and it is currently interpreted as anomalous velocity jumps, positive and negative, of the order of a few mm s−1 observed near the closest approach during the Earth flybys [53, 54]. Several attempts to explain the flyby 75 anomaly have been put forth so far. For example, as far as standard physics is concerned, it was shown that the Rosetta flyby is unlikely due to thermal recoil pressure [55] or to Lorentz forces [56], but might be due to gravitoelectric (contributing up to 10−2 mm s−1)and gravitomagnetic forces (up to 10−5 mm s−1) [57]. Moreover, it was shown that neither the general relativistic Lense- 80 Thirring effect nor a Rindler-type radial uniform acceleration were the cause of the flyby anomaly [23]. Unusual explanations were advanced based on a possible modification of inertia at very low acceleration when Unruh wavelengths exceed the Hubble distance [58]; the elastic and inelastic scattering of ordinary matter with dark matter, although submitted to highly constraints [59, 60]; how 85 Conformal Gravity affects the trajectories of geodesic motion around a rotating spherical object, but are not expected to cause the flyby anomaly [61]. In this paper, we suggest a possible theoretical explanation of the physical process underlying the unexpected orbital-energy change observed during close planetary flybys [22, 62] based on the topological torsion current (TTC) found 90 in a previous work [63]. Anderson et al. [64] proposed a helicity-rotation coupling that is akin to our proposal. However, the anomalous acceleration cannot be explained by means of the helicity-rotation mechanism due to its small magnitude. The TTC was obtained in the framework of a new variational principle based on the fundamental equation of thermodynamics treated as a 95 differential form. That formulation gives a set of two first order differential equations that have the same symplectic structure as classical mechanics, fluid dynamics, and thermodynamics. The procedure can be applied to investigate 4 out-of-equilibrium dynamic systems. From that approach emerges a TTC of the form ijkAj!k, where Aj and !k denote the components of the vector 100 potential (here, the gravitational) and where ! denotes the angular velocity of the accelerated frame. 2. Outlines of the method A standard technique for treating thermodynamical systems on the basis of information-theoretic framework has been developed previously [65, 66, 67, 63]. 105 We can find in technical literature several textbooks that give an overview over the subject, see e.g., Ref. [68, 69, 70, 71, 72, 73]. This work may be applied to a self-gravitating plasma system. The extended mathematical formalism devel- oped to investigate out-of-equilibrium systems in the framework of information theory can be applied to the analysis of the equilibrium and stability of a grav- 110 itational and electromagnetic system (e.g., rotating plasma, or spacecraft in a gravitationally-assisted maneuver). Our method is based on applying Lagrange multipliers to the total entropy of an ensemble of particles.