The Flyby Anomaly and the Effect of a Topological Torsion Current

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 relationship 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

$Revised 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 astrometric 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 anomalous 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 potential acting on the Solar System possibly due to a new hypothetical huge body, Tyche [8, 9]. The phenomenological modification of Newtonian dynamics proposed 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 anomalies, 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 proposed, 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 ijkAjωk, where Aj and ωk denote the components of the vector

100 potential (here, the gravitational) and where ω 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 developed 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. However, we use the fundamental equation of ther- P k modynamics dU = T dS − k Fkdx on diﬀerential forms, considering U and S

115 as 0-forms. As we have shown in a previous work [67] we obtain a set of two first order differential equations that reveal the same formal symplectic structure shared by classical mechanics, fluid mechanics, and thermodynamics. Following the mathematical procedure proposed in Ref. [67] the total entropy of the system S, considered as a formal entity describing an out-of-equilibrium physical system, is given by the following equation:

N X (p(α))2 S = {S(α)[E(α) − − q(α)V (α) + q(α)(A(α) · v(α))− 2m(α) α=1

N X m(α)φ(α)(r) − m(α) φ(α,β)] + (a · p(α) + b · ([r(α) × p(α)])}. (1) β=1 Although it has been argued that S was deﬁned for equilibrium states and had

5 no time dependence of any kind, it might still be possible to describe entropy

120 by some means during the evolution of a physical system. However, if the time evolution of others physical quantities can be made, such as energy E, pressure P and number of particles N, then why not S. As in our previous work [63], regardless of these uncertainties, the explanation proposed here provides a diﬀerent input to move further toward a better understanding of the

125 role of entropy. The conditional extremum points provide the canonical momentum and the dynamical equations of motion of a general physical system in out-of-equilibrium conditions. The two ﬁrst order diﬀerential equations can then be represented in the following form (see Ref. [67]):

¯ ∂p(α) S ≥ 0 (2)

¯ (α) (α) (α) ∂r(α) S = −η∂r(α) U − ηm ∂tv ≥ 0. (3)

Here, η ≡ 1/T is the inverse of the ”temperature” (not being used so far), and

we use condensed notation: ∂p(α) ≡ ∂/ ∂p(α) . Then we obtain a general equation of dynamics for electromagnetic-gravitational systems:

dv ρ = ρE + [J × B] − ∇φ − ∇p + ρ[A × ω]. (4) dt

The last term of Eq. 4 represents the TTC [67]. We stress how A may be considered physically real, even in a gravitational ﬁeld, despite the arbitrariness in its divergence. This force increases the rotational energy of the system, producing a rocket-like rotation eﬀect on a plasma, or the orbital-energy change

130 observed during the close planetary ﬂybys, an issue thoroughly discussed in Ref. [74]. Moreover, the TTC emerges from the universal competition between entropy and energy, each one seeking a diﬀerent equilibrium condition. (This happens in the case of planetary atmospheres, when energy tends to assemble all atmospheric molecules on the surface of the planet, but entropy seeks to spread

6 V F

1 dV= RdI dV= dQ C

dF= Rdv dF=Vdt dF= kdx dp=Fdt

I dQ=idt Q v dx=vdt (x,q) dF= LdI dp= mdv dF= MdQ dp= Ad F=LI p=mv

Figure 1: The missing fourth element of force: following an analogy with the electromagnetic ﬁeld, a new element of force is expected, the topological torsion current (TTC). The ﬁgure uses the standard symbols used for resistors, capacitors, solenoids and memristors.

135 them evenly in all available space). The TTC may be envisaged as the missing force term in the traditional hierarchy of agencies responsible for the motion of matter, as depicted in Fig. 1, and following along the same electromagnetic analogy proposed by Chua [75]. The basic four physical quantities are the electric current i (or speed v), the voltage V (or the force F ), the charge q (or

140 the position x), and the ﬂux-linkage Φ (or momentum p = mv). From a logical point of view, of six possible combinations among these four variables, ﬁve are already well-known. However, the TTC points to the existence of an as yet undiscovered relationship between momentum and angular motion through the agency of a vector potential (see Refs. [75, 76]).

7 145 2.1. Application to the Flyby Anomaly

Implicit in Eq. 4 is the action of the vector potential over a given body, besides the E and B-fields, a term analogue to a rotational electric field. Fig. 2 illustrates the typical planetary flyby a spacecraft makes in the geocentric equatorial frame and the orbital elements, where h is the angular momentum normal

150 to the plane of the orbit and e is the eccentricity vector pointing along the apse line of the arrival hyperbola. Let us apply the new governing equation to the planetary flyby of a given spacecraft of mass m nearby a planet of mass M, as illustrated in Fig. 2 (see, e.g., Ref. [77, 78]). Hence, in cylindrical geometry, and taking into account the TTC effect alone, Eq. 4 becomes the following equation (Fig. 2 shows the Earth flyby geometry): dv m θ = mω A sin I. (5) dt z r

Eq. 5 is written in the geocentric system because that is where the radio tracking data is obtained. Notice that the velocity of the spacecraft relative to the Sun is

given by vsS = vsP + VPS, where vsP is its velocity relative to the planet and

VPS is the velocity of the planet relative to the Sun. But if we consider the term

VPS as time-independent, Eq. 5 provides the azimuthal velocity component of the spacecraft relative to Earth. If we take due care of the retardation of the gravitational ﬁeld, it is appropriate to use the following gravitational vector potential under the (Li´enard-Wiechert) form

G MvsP A(r, t) = 0 . (6) c2 0 vsP ·n | r − r | 1 − c

Here, r is the vector position of the planet (e.g., Earth) and r0 is the vector position of the spacecraft, both in the heliocentric system; n0 is the unit vector 0 0 (r − r )/R, with R =| r − r | (see Fig. 2). We assume that VPS = VPSJ and that the planet moves perpendicularly to the vernal line (the Sun is located on 0 the side of the axis −I) along the J axis (see Fig. 2), and therefore (A·n ) = Ar is the radial component, since what counts in Eq. 6 is the relative velocity

8 between spacecraft and planet. The approach velocity vector vap is expressed in the approach plane (i, j, h) as follows (the unit vector i points along the planet direction of motion):

vap = vapxi + vapyj + vapzh. (7)

The general representation of the spacecraft velocity vector relative to Earth in the direct orthonormal frame is given by the following equation:

vapx = VP + v∞ cos(ω ∓ θ)

vapy = v∞ sin(ω ∓ θ) (8)

vapz = 0.

Here, v∞ is the excess hyperbolic speed of the spacecraft with respect to the planet. We denote by ω⊕ the Earth’s angular velocity of rotation, by R⊕ the Earth’s mean radius, and by G the gravitational constant.

The transit time dt of the spacecraft at the average distance R⊕ (assumed here as the radius of the sphere of influence) from the center of the planet (this approximation is assumed because in general the spacecraft altitude is smaller than R⊕, see also Ref. [22]), and we state that dt = dθR⊕/vθ, where vθ is the azimuthal component of the spacecraft velocity and dθ denotes the angular deflection undergone by the spacecraft during the transit time near the planet. 0 Expanding Eq. 6 to the first order in (vsP · n )/c, we may write Eq. 5 in the following form:

GM Vr GM Vr 0 dvθ = ω⊕ sin I 2 dt + ω⊕ sin I 2 (vsP · n )dt, (9) c R⊕ c R⊕ or in the following form:

GM 2ω⊕r⊕ GM dv∞ = 2ω⊕R⊕ sin I 2 dθ + 2 v∞ sin(ω ∓ θ) sin Idθ (10) 2R⊕c c 2R⊕c

0 We can write the equation in these forms because (vsP ·n ) = v∞ sin(ω ∓θ) and

9 p 2 2 the radial component of the (relative) velocity is Vr = vx + (vy − VP ) = v∞. To simplify Eq. 10 further, we may use the principle of the energy of inertia, which states that the gravitational energy of a spacecraft on the surroundings of the planet must be equal to its energy content according to Einstein’s formula, 2 i.e., the ﬁeld itself carries mass. Therefore, GMm/2R⊕ = mc . As a result of this equivalence, the velocity variation is independent of the mass of the planet, remaining dependent on its radius, angular velocity, and the orbital inclination. To integrate θ instead, we may consider the connection between θ with the declination angle δ using the following trigonometric relationship (see Fig. 2):

sin(ω ∓ θ) sin I = sin δ (11)

In this equation, I denotes the osculating orbital inclination to the equator of date and ω is the osculating argument of the perigee along the orbit from the equator of date. This change allows us to rewrite Eq. 10 in the form of a ﬁrst-order non-linear diﬀerential equation, as follows:

dv ∞ = 2ω R sin I + Kv sin δ(θ) (12) dθ ⊕ ⊕ ∞

Now, vθ = v∞ and δ = δ(θ). It is worth mentioning that the ﬁrst constant term

of Eq. 12 cancels out when calculating the velocity change ∆v∞. Therefore, we obtain the following equation:

Z dv∞ v∞,f ∆v∞ = ln ≈ = K(cos δi − cos δf ). (13) v∞ v∞,i v∞

155 Here, v∞ denotes the azimuthal speed of the spacecraft in a position far-

away from the planetary inﬂuence (R → ∞), K ≡ 2R⊕ω⊕/c is the distance-

independent factor, δi, and δf denote the initial and ﬁnal declination angles on the celestial sphere. Eq. 13 coincides with the heuristic formula proposed by Anderson [64], which is appropriate for spacecrafts below 2000 km of altitude

160 and has been adjusted to high- altitude ﬂybys [79]. According to the present analysis the ﬂyby anomaly may have the follow-

10 Kˆ I - Osculating orbital inclination Z Perigee e v h Spacecraft  R

d - Declination Jˆ

Earth’s equatorial  - Argument of plane perigee a - Right ascension N X Node line Iˆ

g Vernal equinox line

Figure 2: Planetary ﬂyby by a spacecraft in the geocentric equatorial frame and the orbital elements. h is the angular momentum normal to the plane of the orbit and e is the eccentricity vector. I denotes the osculating orbital inclination to the equator of date, and ω is the osculating argument of the periapsis along the orbit from the equator of date.

ing causes: i) a drag effect from the planet by means of a Coriolis-like force that pushes or pulls the spacecraft (different from frame dragging, which is debatable [80]); or ii) a retarded effect from the gravitational field due to rota-

165 tion of the planet. The known result, obtained by using experimental data is −6 ∆v∞/v∞ = K(cos δi −cos δf ), where K = 2ω⊕R⊕/c = 3.099×10 [79, 81, 22]. The dependency on the term sin I indicates that there is no anomalous acceleration when the inclination angle I is equal to zero. This result is consistent with the data of Table 1, which collects the orbital and anomalous dynami-

170 cal parameters of five Earth flybys as presented in Ref. [74]. For example, the Cassini Earth flyby has no registered data because there is no anomaly; in con- trast, when I ∼ 90 ◦, as is the case of NEAR, the variation is boosted to a higher

value ∆v∞ = 13.46 ± 0.13 mm/s. Moreover, the results show that, due to the

11 Table 1: Orbital and anomalous dynamical parameters of five Earth flybys. b is the impact parameter, A is the altitude of the flyby, I is the inclination, α is the right ascension, and δ is the declination of the incoming (i) and outgoing (f) osculating asymptotic velocity vectors. m is the best estimate of the total mass of the spacecraft during the flyby. v∞ is the asymptotic velocity; ∆v∞ is the increase in the asymptotic velocity of the hyperbolic trajectory. Source: Ref. [22, 74]. Quantity Galileo (GEGA1) NEAR Cassini Rosetta b (km/s) 11,261 12,850 8,973 22,680.49 A (km) 956.063 532.485 1171.505 1954.303 I( ◦) 142.9 108.0 25.4 144.9 m (kg) 2497.1 730.4 4612.1 2895.2 α ( ◦) 163.7 240.0 223.7 269.894 δ ( ◦) 2.975 -15.37 -11.16 -28.185 ∆v∞ (mm/s) 3.92 ± 0.08 13.46 ±0.13 ... 1.82 ± 0.05

vectorial nature of the TTC (and its dependence on the inclination angle), the

175 anomaly can either increase or decrease depending on whether the spacecraft encounters Earth on the leading or trailing side of its orbital path.

3. Conclusion

We may conclude that the variational method proposed in Ref. [63] consti- tutes a powerful alternative approach to tackle problems in the frame of gravita-

180 tional and/or electromagnetic rotating systems. The emergence of a new force term - the TTC - offers a simple explanation for the flyby anomaly; namely, that it results from a combined slingshot effect (which is not identifiable to frame-dragging) with retardation effects due to the noninstantaneous character of the gravitational force. In principle, the physical mechanism proposed in

185 this Letter should be applicable to other systems as well, such as closed orbits and the other anomalies referred in the Introduction, tasks to be undertaken in future work. In addition, the TTC may be the missing fourth element of force if we consider the traditional hierarchy of agencies responsible for the motion of matter

190 (see Fig. 1) and the electromagnetic analogy proposed by Chua [75]. Based upon the same logical and axiomatic point of view, we may establish an op- erational relationship between linear and angular motion. Achieving a deeper

12 comprehension of the trajectory of the Near-Earth Objects, such as asteroids and comets, requires a change in the standard assumptions. Understanding the

195 TTC contribution to the gravitational force will be instrumental when accessing their trajectories.

The author thanks the three anonymous referees for their useful comments that greatly improved the quality of the manuscript and gratefully acknowl- edges partial ﬁnancial support by the International Space Science Institute

200 (ISSI-Bern) as a visiting scientist, expressing special thanks to Professor Roger- Maurice Bonnet and Dr. Maurizio Falanga.

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