1

1

The Flyby Anomaly and the Effect of a TopologicalTorsion Current

Mario J. Pinheiro1

aDepartment of Physics, Instituto Superior Tecnico - 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 and/or electromagnetic system and provides a modified

dynamical equation of motion from where it emerges a so-far unforseen topo-

logical torsion current (TTC) [Mario J. Pinheiro (2013) ’A Variational 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 an unforeseen 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. Introduction1

Flyby (or swing-by, gravitational slingshot, or gravity assist maneuver) is2

a well-known method in interplanetary spaceflight to alter the path and the3

speed of a spacecraft using the gravity of a planet or other astronomical object4

URL: http://web.ist.utl.pt/d2493/ (Mario J. Pinheiro)1mpinheiro@tecnico.ulisboa.pt

Preprint submitted to Elsevier August 12, 2014

(see, e.g., Ref. [1]). The rescue of the Apollo crew in 1970 was the first flyby5

maneuver ever did, using the Lunar flyby [2].6

But the flyby anomaly is one among other, possibly related, several astro-7

metric anomalies that are referred in the technical literature, such as the change8

of the solar mass over time M� (changes that result from a balance between the9

mass loss due to radiation and solar wind compensated by falling materials con-10

tained in comets, rocks and asteroids) leading to the observation of a decrease of11

the heliocentric gravitation constant per year GM�/GM� = (−5.0±4.1).10−1412

per year and a changing of the astronomical unit by approximately 10 m per cen-13

tury [3]. Quite surprisingly, dark matter does not have a gravitational influence14

in the solar system because its density is very low [4]. The angular momentum15

of the Sun seems to be smaller than expected (S� ≤ 0.95 × 1041 kg m2 s−1)16

unless the prediction of the gravitomagnetic perihelion precession of Mercury in17

the frame of the best theory of the gravitational interaction which is the Ein-18

steinian General Theory of Relativity is wrong [5, 6]. The anomalous behavior of19

the Saturnian perihelion cannot be explained in the framework of the standard20

Newtonian and Einsteinian General Theory of Relativity [7], also suggesting21

the need of new physics or the effect of an external tidal potential acting on the22

Solar System or a new hypothetical huge body, Tyche [8, 9]. The phenomeno-23

logical modification of Newtonian dynamics proposed by MOND doesn’t offer a24

satisfying explanation for Cassini spacecraft anomaly [8]. The Faint Young Sun25

Paradox [10, 11, 12] can possibly be accommodated within a certain general26

class of gravitational theories with nonminimal coupling between metric and27

matter predicting a secular variation of the Earth heliocentric distance [13, 14].28

Recent analysis of a Lunar Laser Ranging data record revealed an anomalous29

increase of the eccentricity rate of the lunar orbit [15, 16, 14, 17]. This effect30

is not related to a possible change of the speed of light [16] or some dissipation31

at the lunar core and mantle [15], but possibly non-tidal explanations can be32

viable [17].33

However, astrometric data points to the existence of at least four unexplained34

anomalies, from the small and constant Doppler frequency drift shown by the35

3

radio-metric data from Pioneer 10/11, which can be interpreted as a uniform36

acceleration of aP = (8.74 ± 1.33)10−8 cm/s2 towards the Sun found in the37

data of both spacecraft when they were at a distance of 20 au away from the38

Sun [18, 19, 20, 21, 22, 23] to the disturbing observation that a number of39

satellites in Earth flyby have undergone mysterious energy changes [22]. This40

effect is essentially a slight departure from Newtonian acceleration (see also41

Ref. [24] for an overview of unexplained phenomena within our Solar system42

and in the universe) having been shown in different number of ways [18, 20]43

that it is not a real gravitational phenomena that would certainly have impacted44

also other major bodies of the solar system. The possibility that this uniform45

Sunward acceleration, as the one experienced by the Pioneer spacecraft, might46

have a gravitational nature, was shown to be erroneous [25, 26, 27, 28], not even47

being able to affect the motion of the outer planets of the Solar system [29, 30].48

Rindler-type extra-acceleration on test particles were ruled out altogether since49

it would affect the main features of the Oort cloud [31] and exotic physics is50

most probably not affecting Pioneer spacecraft trajectories [32].51

The possibility the the Pioneer anomaly might have a different non-gravitational52

origin, like a recoil force associated with an anisotropic emission of thermal ra-53

diation off the spacecraft was proposed [33, 34, 35, 36, 37, 38, 39]. As referred54

before, a secular change in the astronomical unit (au) d(au)dt = 7±2 m cy−1 [40]55

was reported [41, 40, 42] and several explanations were proposed, among them,56

the change in the moment of inertia of the Sun due to radiative mass loss [43]57

but the possible variation of the dark matter density was ruled out [44]. The58

huge importance of the problem and the uncertainties related to the causes of59

its variation lead to the proposal of fixing the value of au [45, 46, 47]. There is60

other proposed explanations for this effect, among them we may refer: an adi-61

abatic acceleration of light due to an adiabatic decreasing of the permeability62

and permittivity of empty space [48]; the dilaton-like Jordan-Brans-Dicke scalar63

field as the source of dark energy and giving rise to a new term of force with64

magnitude aP = Fr/m = −c2/RH (RH is the Hubble scale), see Ref. [49]; light65

speed anisotropy [50] based on Lorentz space-time interpretation and resorting66

4

from the earlier measurement od D. C. Miller (see also Ref. [51] which gives an67

interesting reformulation of special theory of relativity); a computer modeling68

technique called Phong reflection model [52] may apparently explains the effect69

as mainly due to the heat reflected from the main compartment, but it still70

needs confirmation.71

The flyby anomaly appears as a shift in the Doppler data of Earth-flybys72

of several spacecrafts and it is currently interpreted as an anomalous velocity73

jumps, positive and negative, of the order of a few mm s−1 observed near closest74

approach during the Earth flybys [53, 54]. Several attempts to explain the flyby75

anomaly have been put forth so far. For example, as far as standard physics76

is concerned, it was shown that Rosetta flyby is unlikely due to thermal recoil77

pressure [55], it is not due to Lorentz forces [56], might be due to gravitoelec-78

tric (contributing up to 10−2 mm s−1)and gravitomagnetic forces (up to 10−579

mm s−1) [57]. It was shown that the general relativistic Lense-Thirring effect80

or a Rindler-type radial uniform acceleration were not the cause of the flyby81

anomaly [23]. Exotic explanations were advanced based on a possible modifi-82

cation of inertia at very low acceleration when Unruh wavelengths exceed the83

Hubble distance [58]; the elastic and inelastic scattering of ordinary matter with84

dark matter, although submitted to highly constraints [59, 60]; how Conformal85

Gravity affects the trajectories of geodesic motion around a rotating spherical86

object, but are not expected to cause the flyby origin [61].87

In this paper, we suggest a possible theoretical explanation of the physical88

process underlying the unexpected orbital-energy change observed during the89

close planetary flybys [22, 62] based on the topological torsion current (TTC)90

found in a previous work [63]. Anderson et al. [64] proposed an helicity-rotation91

coupling that is more akin to our proposal. However, the anomalous acceleration92

cannot be explained by means of their mechanism due to its small magnitude.93

The topological torsion current was obtained in the framework of a new varia-94

tional principle based on the fundamental equation of thermodynamics treated95

as a differential form. That formulation gives a set of two first order differen-96

tial equations that have the same symplectic structure as classical mechanics,97

5

fluid dynamics and thermodynamics. That procedure can be applied to inves-98

tigate out-of-equilibrium dynamic systems and from that approach it emerges99

a topological torsion current of the form εijkAjωk, where Aj and ωk denote100

the components of the vector potential (here, the gravitational) and where ω101

denotes the angular velocity of the accelerated frame.102

2. Outlines of the method103

A standard technique for treating thermodynamical systems on the basis of104

information-theoretic framework has been developed previously [65, 66, 67, 63].105

We can find in technical literature several textbooks that give an overview over106

the subject, see e.g., Ref. [68, 69, 70, 71, 72, 73]. The referred work may be107

applied to a self-gravitating plasma system, and the extended mathematical108

formalism developed to investigate out-of-equilibrium systems in the framework109

of the information theory, can be applied for the analysis of the equilibrium and110

stability of a gravitational and electromagnetic system (e.g., rotating plasma,111

or spacecraft in a gravitationally-assisted manoeuver).112

Our method is fundamentally based on the method of Lagrange multipliers113

applied to the total entropy of an ensemble of particles. However, we use the114

fundamental equation of thermodynamics dU = TdS−∑k Fkdx

k on differential115

forms, considering U and S as 0-forms. As we have shown in a previous work [67]116

we obtain a set of two first order differential equations that reveal the same117

formal symplectic structure shared by classical mechanics, fluid mechanics and118

thermodynamics.119

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

S =

N∑α=1

{S(α)[E(α) − (p(α))2

2m(α)− q(α)V (α) + q(α)(A(α) · v(α))−

6

m(α)φ(α)(r)−m(α)N∑β=1

φ(α,β)] + (a · p(α) + b · ([r(α) × p(α)])}. (1)

Although it has been argued that S was defined for equilibrium states and had120

no time dependence of any kind, one might think that it must be possible to121

describe entropy by some means during the evolution of a physical system. But122

if the time evolution of others physical quantities can be made, like energy E,123

pressure P and number of particles N , then why not S. As in our previous124

work [63], regardless of these uncertainties, the explanation proposed here pro-125

vides a different input to move further toward a better understanding of the126

role of entropy.127

The conditional extremum points give the canonical momentum and the

dynamical equations of motion of a general physical system in out-of-equilibrium

conditions. Then the two first order differential equations can be represented in

the form (see Ref. [67]):

∂p(α) S ≥ 0 (2)

∂r(α) S = −η∂r(α)U (α) − ηm(α)∂tv(α) ≥ 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:

ρdv

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

The last term of Eq. 4 represents the topological torsion current [67] (TTC)128

and we may stress how A may be considered physically real, even in the frame129

of the gravitational field, despite eventually the arbitrariness in its divergence.130

This force does work in order to increase the rotational energy of the system,131

producing a rocket-like rotation effect on a plasma, or the orbital-energy change132

7

observed during the close planetary flybys, an issue thoroughly discussed in133

Ref. [74]. Moreover, the topological torsion current emerge from the universal134

competition between entropy and energy, each one seeking a different equilib-135

rium condition (this happens in the case of planetary atmospheres, when energy136

tends to assemble all atmospheric molecules on the surface of the planet, but137

entropy seeks to spread them evenly in all available space). This TTC may138

be envisaged as the missing force term in the traditional hierarchy of agencies139

responsible for the motion of matter, as depicted in Fig. 1, and following along140

the same electromagnetic analogy proposed by Chua [75]. The basic four phys-141

ical quantities are the electric current i (or speed v), the voltage V (or the142

force F ), the charge q (or the position x), and the flux-linkage Φ (or momen-143

tum p = mv). Under the logical point of view, from six possible combinations144

among these four variables, five are already well-known. However, the TTC145

points to the existence of a so-far unforeseen relationship between momentum146

and angular motion through the agency of a vector potential (see Refs. [75, 76]).147

2.1. Application to the Flyby Anomaly148

It is implicit into Eq. 4 the action of the vector potential over a given body,149

besides the E and B-fields, a term analogue to a rotational electric field. Fig. 2150

illustrates the typical planetary flyby by a spacecraft in the geocentric equatorial151

frame and the orbital elements, where h is the angular momentum normal to152

the plane of the orbit and e is the eccentricity vector pointing along the apse153

line of the arrival hyperbola.154

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 (Fig. 2 shows the Earth flyby geometry):

mdvθdt

= mωzAr sin I. (5)

The above Eq. 5 is written in the geocentric system since it is where the radio

8

V F

I v

Q (x,q)

F=LI p=mv

dQ=idt dx=vdt

dF=Vdt dp=Fdt

dV RdI=

dF Rdv=

1dV dQ

C=

dF kdx=

d LdIF =

dp mdv= d MdQF =

dp Ad=

Figure 1: The missing fourth element of force: following an analogy with the electromagneticfield, it is expected a new element of force, the topological torsion current. It is shown thestandard symbols used for resistors, capacitors, solenoids and memristors.

9

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 time-independent, Eq. 5 gives at the end the azimuthal

velocity component of the spacecraft relative to Earth. Then, if we take due

care of the retardation of the gravitational field, it is appropriate to use the

gravitational vector potential under the (Lienard-Wiechert) form

A(r, t) =G

c2MvsP

| r− r′ |(1− vsP ·n′

c

) . (6)

Here, r is the vector position of the planet (e.g., Earth) and r′ is the vector

position of the spacecraft, both in the heliocentric system; n′ is the unit vector

(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 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

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)

and the general representation of the spacecraft velocity vector relative to Earth

in the direct orthonormal frame is given by

vapx = VP + v∞ cos(ω ∓ θ)

vapy = v∞ sin(ω ∓ θ)

vapz = 0.

(8)

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, R⊕ the

Earth’s mean radius, G the gravitational constant. The transit time dt of the

10

spacecraft at the average distance R⊕ (assumed here the radius of the sphere of

influence) from the center of the planet (we assume this approximation, since

in general the spacecraft altitude is rather smaller than R⊕, see also Ref. [22]),

and we put dt = dθR⊕/vθ, where vθ is the azimuthal component of the space-

craft velocity, and dθ denotes the angular deflexion undergone by the spacecraft

during the transit time nearby the planet. Expanding Eq. 6 to first order in

(vsP · n′)/c, we may write Eq. 5 under the form:

dvθ = ω⊕ sin IGM

c2VrR⊕

dt+ ω⊕ sin IGM

c2VrR⊕

(vsP · n′)dt, (9)

or,

dv∞ = 2ω⊕R⊕ sin IGM

2R⊕c2dθ +

2ω⊕r⊕c

GM

2R⊕c2v∞ sin(ω ∓ θ) sin Idθ (10)

since (vsP ·n′) = v∞ sin(ω∓θ) and the radial component of the (relative) velocity

is Vr =√v2x + (vy − VP )2 = v∞. In order to simplify further Eq. 10 we may use

now the principle of the energy of inertia, which states that the gravitational

energy of the spacecraft on the surroundings of the planet must be equal to its

energy content according to Einstein formula, e.g., the field itself carries mass,

and hence GMm/2R⊕ = mc2. As a result from this equivalence, the velocity

variation is independent of the mass of the planet, remaining dependent of its

radius, angular velocity and the orbital inclination. Instead to integrate in θ we

may consider the connection between θ with the declination angle δ using the

trigonometric relationship (see Fig. 2):

sin(ω ∓ θ) sin I = sin δ, (11)

where I denotes the osculating orbital inclination to the equator of date, ω

is the osculating argument of the perigee along the orbit from the equator of

date. This change allows to rewrite Eq. 10 in the form of a first-order non-linear

11

differential equation

dv∞dθ

= 2ω⊕R⊕ sin I +Kv∞ sin δ(θ), (12)

where we have now put vθ = v∞ and noting that 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:

∫dv∞v∞

= lnv∞,fv∞,i

≈ ∆v∞v∞

= K(cos δi − cos δf ). (13)

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

away from the planetary influence (R → ∞), K ≡ 2R⊕ω⊕/c is the distance-156

independent factor, δi and δf denote the initial and final declination angles on157

the celestial sphere. Eq. 13 coincides with the heuristic formula proposed by158

Anderson [64], fitting well for spacecrafts below 2000 km of altitude and has159

been so far adjusted to high altitudes flyby [79].160

According to the present analysis the flyby anomaly may have the following161

causes: i) a drag effect from the planet by means of a Coriolis-like force that push162

or pull the spacecraft (different from frame dragging, which is debatable [80]);163

ii) a retarded effect from the gravitational field due to rotation of the planet.164

The known result, obtained by fitting with experimental data is ∆v∞/v∞ =165

K(cos δi − cos δf ), where K = 2ω⊕R⊕/c = 3.099× 10−6 [79, 81, 22].166

The dependency on the term sin I indicates that there is no anomalous ac-167

celeration when the inclination angle I is equal to zero. This result is consistent168

with the data of Table 1 collecting the orbital and anomalous dynamical pa-169

rameters of five Earth flybys as presented in Ref. [74]. For example, Cassini170

Earth flyby has no registered data for it just because there is no anomaly; by171

the contrary, when I ∼ 90 ◦, as is the case of NEAR, the variation boost to a172

higher value ∆v∞ = 13.46 ± 0.13 mm/s. Moreover, from the results obtained173

we see why, due to the vectorial nature of the topological torsion current (and174

its dependency on the inclination angle), the anomaly can either increase or175

12

X

g

Y

N

Earth’s equatorial

plane

Node line

Vernal equinox line

R

vh

Z e

Spacecraft

Perigee

d

a - Right ascension

- Declination

I

J

K

- Argument of

perigee

I - Osculating orbital

inclination

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

13

Table 1: Orbital and anomalous dynamical parameters of five Earth flybys. b is the impactparameter, A is the altitude of the flyby, I is the inclination, α is the right ascension, δ is thedeclination of the incoming (i) and outgoing (f) osculating asymptotic velocity vectors. m isthe best estimate of the total mass of the spacecraft during the flyby. v∞ is the asymptoticvelocity; ∆v∞ is the increase in the asymptotic velocity of the hyperbolic trajectory. Source:Ref. [22, 74].

Quantity Galileo (GEGA1) NEAR Cassini Rosettab (km/s) 11,261 12,850 8,973 22,680.49A (km) 956.063 532.485 1171.505 1954.303I ( ◦) 142.9 108.0 25.4 144.9m (kg) 2497.1 730.4 4612.1 2895.2α ( ◦) 163.7 240.0 223.7 269.894δ ( ◦) 2.975 -15.37 -11.16 -28.185∆v∞ (mm/s) 3.92 ± 0.08 13.46 ±0.13 ... 1.82 ± 0.05

decrease depending if the spacecraft encounters Earth on the leading or trailing176

side of its orbital path.177

3. Conclusion178

We may conclude from the above that the variational method proposed in179

Ref. [63] constitutes a powerful alternative approach to tackle problems in the180

frame of gravitational and/or electromagnetic rotating systems. The emergence181

of a new force term - the topological torsion current - offers a simple explanation182

for the flyby anomaly, in fact resulting from a combined slingshot effect (which183

is not identifiable to frame-dragging) with retardation effects due to the non-184

instantaneous character of the gravitational force. In principle, the physical185

mechanism proposed in this Letter should be applicable to other systems as186

well, like closed orbits and the other anomalies referred in the Introduction, a187

task to be undertaken in future work.188

In addition, the TTC may be well the missing fourth element of force if we189

consider the traditional hierarchy of agencies responsible for the motion of mat-190

ter (see Fig. 1) and the electromagnetic analogy proposed by Chua [75]. Based191

upon the same logical and axiomatic point of view we may establish an opera-192

tional relationship between linear and angular motion. A deeper understanding193

of the trajectory of the Near-Earth Objects, like asteroids and comets, raising194

14

significant concerns, needs a change in the standard assumptions and certainly195

a deeper understanding of the TTC contribution to the gravitational force will196

be instrumental when accessing their trajectories.197

The author thanks the three anonymous referees for their useful comments198

that greatly improved the quality of the manuscript and gratefully acknowl-199

edges partial financial support by the International Space Science Institute200

(ISSI-Bern) as a visiting scientist, expressing special thanks to Professor Roger201

Maurice-Bonnet and Dr. Maurizio Falanga.202

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