Dark matter based model for Flyby anomaly
The so called flyby anomaly provides a test for any theory of gravitation, quantal or not. I have already earlier discussed a model of this anomaly based on dark matter located either at spherical shell or tube around Earth's orbit. The recent data (see this and this ) allowed to fix the model to a tube around the orbit of Earth deformed by gravimagnetic force of Earth to the direction of equatorial plane of Earth.
1. Flyby anomaly
Flyby mechanism used to accelerate spacecrafts is a genuine three body effect involving Sun, planet, and the spacecraft. Planets are rotating around sun in an anticlockwise manner and when the spacecraft arrives from the right hand side, it is attracted by a planet and is deflected in an anticlockwise manner and planet gains energy as measured with respect to solar center of mass system. The energy originates from the rotational motion of the planet. If the spacecraft arrives from the left, it loses energy. What happens is analyzed the above linked article using an approximately conserved quantity known as Jacobi's integral
J= e ω e_{z} · r× v.
Here e is total energy per mass for the spacecraft, ω is the angular velocity of the planet, e_{z} is a unit vector normal to the planet's rotational plane, and various quantities are with respect to solar cm system.
This as such is not anomalous and flyby effect is used to accelerate spacecrafts. For instance, Pioneer 11 was accelerated in the gravitational field of Jupiter to a more energetic elliptic orbit directed to Saturn ad the encounter with Saturn led to a hyperbolic orbit leading out from solar system.
Consider now the anomaly. The energy of the spacecraft in planetspacecraft cm system is predicted to be conserved in the encounter. Intuitively this seems obvious since the time and length scales of the collision are so short as compared to those associated with the interaction with Sun that the gravitational field of Sun does not vary appreciably in the collision region. Surprisingly, it turned out that this conservation law does not hold true in Earth flybys. Furthermore, irrespective of whether the total energy with respect to solar cm system increases or decreases, the energy in cm system increases during flyby in the cases considered.
Five Earth flybys have been studied: GalileoI, NEAR, Rosetta, Cassina, and Messenger and the article of Anderson and collaborators gives a nice quantitative summary of the findings and of the basic theoretical notions. Among other things the tables of the article give the deviation δe_{g,S} of the energy gain per mass in the solar cm system from the predicted gain. The anomalous energy gain in rest Earth cm system is δe_{E}≈ v·δv and allows to deduce the change in velocity. The general order of magnitude is δv/v≈ 10^{6} for GalileoI, NEAR and Rosetta but consistent with zero for Cassini and Messenger. For instance, for Galileo I one has v_{inf,S}= 8.949 km/s and δv_{ inf,S}= 3.92+/ .08 mm/s in solar cm system.
Many explanations for the effect can be imagined but dark matter is the most obvious candidate in TGD framework. The model for the Bohr quantization of planetary orbits assumes that planets are concentrations of the visible matter around dark matter structures. These structures could be tubular structures around the orbit or a nearly spherical shell containing the orbit. The contribution of the dark matter to the gravitational potential increases the effective solar mass M_{eff,S}. This of course cannot explain the acceleration anomaly which has constant value. One can also consider dark matter rings associated with planets and perhaps even Moon's orbit is an obvious candidate now. It turns out that the tube associated with Earth's orbit and deformed by Earth's presence to equatorial plane of Earth explains qualitatively the known facts.
2. Dark matter at the orbit of Earth?
The almost working model is based on dark matter on the orbit of Earth. One can estimate the change of the kinetic energy in the following manner.
 Assume that the the orbit is not modified at all in the lowest order approximation and estimate the kinetic energy gained as the work done by the force caused by the dark matter on the spacecraft.
ΔE/m= Gdρ_{dark}/dl × ∫_{γE}dl _{E}∫_{γS} dr_{S}• r_{SE }/r_{SE}^{3} ,
r_{SE}== r_{S}r_{E} .
Here γ_{S} denotes the portion of the orbit of spacecraft during which the effect is noticeable and γ_{E} denotes the orbit of Earth.
This expression can be simplified by performing the integration with respect to r_{S} so that one obtains the difference of gravitational potential created by the dark matter tube at the initial and final points of the portion of γ_{S}:
ΔE/m= V(r_{S,f})V(r_{S,i}),
V(r_{S})=G×(dρ_{dark}/dl)×∫_{γE}dl _{E} /r_{SE}
 Use the standard approximation (briefly described in (see this)) in which the orbit of the spacecraft consists of three parts joined continuously together: the initial Kepler orbit around Sun, the piece of orbit which can be approximate with a hyperbolic orbit around Earth, and the final Kepler orbit around Sun. The piece of the hyperbolic orbit can be chosen to belong inside the so called sphere of influence, whose radius r is given in terms of the distance R of planet from Sun by the Roche limit r/R= (3m/M_{Sun})^{2/5}. γ_{S} could be in the first approximation taken to correspond to this portion of the orbit of spacecraft.
 The explicit expression for the hyperbolic orbit can be obtained by using the conservation of energy and angular momentum and reads as
u=r_{s}/r= 2GM/r= (u_{0}^{2}/2v_{0}^{2})×(1+X^{1/2}],
X=1+4u _{0}^{2}×v_{∞}^{2}v_{0}^{2}/sin^{2}(φ),
u_{0}== r_{s}/a , v×r== vr ×sin(φ) .
The unit c=1 is used to simplify the formulas. r_{s} denotes Schwartschild radius and v_{∞} the asymptotic velocity. v_{0} and a are the velocity and distance at closest approach and the conserved angular momentum is given by L/m= v_{0 }a. In the situation considered value of r_{S} is around 1 cm, the value of a around 10^{7} m and the value of v_{∞} of order 10 km/s so that the approximation
u ≈ u_{0}× (v_{∞}/v_{0})×sin(φ)
is good even at the distance of closest approach. Recall that the parameters characterizing the orbit are the distance a of the closest approach, impact parameter b, and the angle 2θ characterizing the angle between the two straight lines forming the asymptotes of the hyperbolic orbit in the orbital plane P_{E}.
Consider first some conclusions that one can make from this model.
 Simple geometric considerations demonstrate that the acceleration in the region between Earth's orbit and the part of orbit of spacecraft for which the distance from Sun is larger than that of Earth is towards Sun. Hence the distance of the spacecraft from Earth tends to decrease and the kinetic energy increases. In fact, one could also choose the portion of γ_{S} to be this portion of the spacecraft's orbit.
 ΔE depends on the relative orientation of the normal n_{S} of the the orbital plane P_{E} of spacecraft with respect to normal n_{O} the orbital plane P_{O} of Earth. The orientation can be characterized by two angles. The first angle could be the direction angle Θ of the position vector of the nearest point of spacecraft's orbit with respect to cm system. Second angle, call it Φ, could characterize the rotation of the orbital plane of spacecraft from the standard orientation in which orbital plane and spacecraft's plane are orthogonal. Besides this ΔE depends on the dynamical parameters of the hyperbolic orbit of spacecraft given by the conserved energy E_{tot} =E_{∞} and angular momentum or equivalently by the asymptotic velocity v_{ ∞} and impact parameter b.
 Since the potential associated with the closed loop defined by Earth's orbit is expected to resemble locally that of a straight string one expects that the potential varies slowly as a function of r_{S} and that ΔE depends weakly on the parameters of the orbit.
The most recent report (see this ) provides additional information about the situation.
 ΔE is reported to be proportional to the total orbital energy E_{∞ }/m of the spacecraft. Naively one would expect (E_{∞}/m)^{1/2} behavior coming from the proportionality ΔE to 1/r. Actually a slower logarithmic behavior is expected since a potential of a linear structure is in question.
 ΔE depends on the initial and final angles θ_{i} and θ _{f} between the velocity v of the spacecraft with respect to the normal n_{E} of the equatorial plane P_{E} or Earth and the authors are able to give an empirical formula for the energy increment. The angle between P_{E} and P_{ O} is 23.4 degrees. One might hope that the formula could be written also in terms of the angle between v and the normal n_{O} of the orbital plane. For θ_{i} ≈ θ_{f} the effect is known to be very small. A particular example corresponds to a situation in which one has θ_{i}=32 degrees and θ_{f }=31 degrees. Obviously the P_{O}≈ P_{E} approximation cannot hold true. Needless to say, also the model based on spherical shell of dark matter fails.
3. Is the tube containing the dark matter deformed locally into the equatorial plane?
The previous model works qualitatively if the interaction of Earth and flux tube around Earth's orbit containing the dark matter modifies the shape of the tube locally so that the portion of the tube contributing to the anomaly lies in a good approximation in P_{E} rather than P _{O}. In this case the minimum value of the distance r_{ES} between γ _{E} and γ_{S} is maximal for the symmetric situation with θ_{i }=θ_{f} and the effect is minimal. In an asymmetric situation the minimum value of r_{ES} decreases and the size of the effect increases. Hence the model works at least qualitatively of the motion of Earth induces a moving deformation of the dark matter tube to P_{E}. With this assumption one can write ΔE in a physically rather transparent form showing that it is consistent with the basic empirical findings.
 By using linear superposition one can write the potential as sum of a potential associated with a tube associated with Earths orbit plus the potential associated with the deformed part minus the potential associated with corresponding nondeformed portion of Earth's orbit:
ΔE/m= V(r_{S,f})V(r_{S,i}) ,
V(r_{S})=G×(dρ_{dark}/dl)Z(r_{S}) ,
Z(r_{S})= X(γ_{orb};r_{S})+ X(γ_{d};r_{S}) X(γ_{nd};r_{S}) ,
X(γ_{i};r_{S}) = ∫_{γi}dl/r_{Si}.
Here the subscripts "orb", "d" and "nd" refer to the entire orbit of Earth, to its deformed part, and corresponding nondeformed part. The entire orbit is analogous to a potential of straight string and is expected to give a slowly varying term which is however nonvanishing in the asymmetric situation. The difference of deformed and nondeformed parts gives at large distances dipole type potential behaving like 1/r^{2} and thus being proportional to v _{∞}^{2} by the above expression for the u=r_{s}/r. The fact that ΔE is proportional to v_{∞}^{2} suggests that dipole approximation is good.
 One can therefore parameterize ΔE as
ΔE/m= V(r_{S,f})V(r_{S,i}) , V(r_{S})=G×(dρ _{dark}/dl)×Z ,
Z(r_{S})= X(γ_{orb};r_{S})+ d×cos(Θ)/r_{S} ^{2},
where Θ is the angle between r and the dipole d, which now has dimension of length. The direction of the dipole is in the first approximation in the equatorial plane and and directed orthogonal to the Earth's orbit.
Consider now the properties of ΔE.
 In a situation symmetric with respect to the equator E_{d} vanishes but E_{nd} is nonvanishing which gives as a result potential difference associated with entire Earth's orbit minus the part of orbit contributing to the effect so that the result is by the definition of the approximation very small.
 As already noticed, dipole field like behavior that the large contribution to the potential is proportional to the conserved total energy v_{0}^{2}/2 at the limit of large kinetic energy.
 From the fact that potential difference is in question it follows that the expression for the energy gain is the difference of parameters characterizing the initial and final situations. This conforms qualitatively with the observation that this kind of difference indeed appears in the empirical fit.
1/r^{2}factor is also proportional to sin^{2}(φ) which by the symmetry of the situation is expected to be same for initial and final situation. Furthermore, ΔE is proportional to the difference of the parameter cos(Θ_{f})cos(Θ_{i}) and this should correspond to the reported behavior. Note that the result vanishes for the symmetric situation in accordance with the empirical findings.
To sum up, it seems that the qualitative properties of ΔE are indeed consistent with the empirical findings. The detailed fit of the formula of the recent paper should allow to fix the shape of the deformed part of the orbit.
4. What induces the deformation?
Authors suggest that the Earth's rotation is somehow involved with the effect. The first thing to notice is that the gravimagnetic field of Earth, call it B_{E}, predicted by General Relativity is quite too weak to explain the effect as a gravimagnetic force on spacecraft and fails also to explain the fact that energy increases always. GravitoLorentz force does not do any work so that the total energy is conserved and ΔE=ΔV=grad V.•Δ r holds true, where Δr is the deflection caused by the gravimagnetic field on the orbit during flyby. Since Δr is linear in v, ΔE changes sign as the velocity of spacecraft changes sign so that this option fails in several manners.
Gravimagnetic force of Earth could be however involved but in a different manner.
 The gravimagnetic force between Earth and flux tube containing the dark matter could explain this deformation as a kind of frame drag effect: dark matter would tend to follow the spinning of Earth. If the dark matter inside the tube is at rest in the rest frame of Sun (this is not a necessary assumption), it moves with respect to Earth with a velocity v=v_{E }, where v_{E} is the orbital velocity of Earth. If the tube is thin, the gravitoLorentz force experienced by dark matter equals in the first approximation to F=v_{E }× B_{E} with B_{E} evaluated at the axis of the tube. TGD based model for B_{E} (see this) does not allow B_{E} to be a dipole field. B_{E} has only the component B^{θ} and the magnitude of this component relates by a factor 1/sin(θ) to the corresponding component of the dipole field and becomes therefore very strong as one approaches poles. The consistency with the existing experimental data requires that B_{E} at equator is very nearly equal to the strength of the dipole field. The magnitude of B_{E} and thus of F is minimal when the deformation of the tube is in P_{E}, and the deformation occurs very naturally into P _{E} since the nongravitational forces associated with the dark matter tube must compensate a minimal gravitational force in dynamical equilibrium.
 B^{θ}_{E} at equator is in the direction of the spin velocity ω of the Earth. The direction of v_{E} varies. It is convenient to consider the situation in the rest system of Sun using Cartesian coordinates for which the orbital plane of Earth corresponds to (x,y) plane with x and yaxis in the direction of semiminor and semimajor axes of the Earth's orbit. The corresponding spherical coordinates are defined in an obvious manner. v_{E} is parallel to the tangent vector e_{φ}(t)=sin (Ωt) e_{x}+ cos(Ωt)e_{y} of the Earth's orbit. The direction of B _{E} at equator is parallel to ω and can be parameterized as e_{ω}= cos(θ) e_{z}+ sin(θ)(cos(α) e_{x}+sin(α)e_{y}). F is parallel to the vector cos(θ) e_{ρ}(t) + sin(θ)cos(Ωtα)e_{z}, where e_{ρ}(t) is the unit vector directed from Sun to Earth. The dominant component is directed to Sun.
Happy note added: The price of "/public_html/articles/ of PRL are really dirty, 25 dollars per article. Hence I decided to take the risk and put the prediction on blog without knowing whether it is correct. Today I got the article in email. The prediction for the increment of kinetic energy was correct! Champagne for that! Flyby will be for the TGD based view about dark matter what the shift of perihelion of Mercury was for General Relativity! Let us now cross our fingers and hope that it would not take too many decades for this message to diffuse through the cognitive immune system of the super string hegemony.
For TGD based view about astrophysics see the chapter TGD and Astrophysics .
