1. Flyby anomaly
Fly-by mechanism used to accelerate space-crafts is a genuine three body effect involving Sun, planet, and the space-craft. Planets are rotating around sun in an anticlockwise manner and when the space-craft 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 space-craft 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 ez · r× v.
Here e is total energy per mass for the space-craft, ω is the angular velocity of the planet, ez 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 space-crafts. 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 space-craft in planet-space-craft 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: Galileo-I, 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 δeg,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 δeE≈ v·δv and allows to deduce the change in velocity. The general order of magnitude is δv/v≈ 10-6 for Galileo-I, NEAR and Rosetta but consistent with zero for Cassini and Messenger. For instance, for Galileo I one has vinf,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 Meff,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 space-craft.
ΔE/m= -Gdρdark/dl × ∫γEdl E∫γS drS• rSE /rSE3 ,
rSE== rS-rE .
Here γS denotes the portion of the orbit of space-craft during which the effect is noticeable and γE denotes the orbit of Earth.
This expression can be simplified by performing the integration with respect to rS 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 γS: ΔE/m= V(rS,f)-V(rS,i),
V(rS)=-G×(dρdark/dl)×∫γEdl E /rSE
- 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/MSun)2/5. γS 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=rs/r= 2GM/r= (u02/2v02)×(1+X1/2],
u0== rs/a , |v×r|== vr ×sin(φ) .
The unit c=1 is used to simplify the formulas. rs denotes Schwartschild radius and v∞ the asymptotic velocity. v0 and a are the velocity and distance at closest approach and the conserved angular momentum is given by L/m= v0 a. In the situation considered value of rS is around 1 cm, the value of a around 107 m and the value of v∞ of order 10 km/s so that the approximation
u ≈ u0× (v∞/v0)×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 PE.
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 γS to be this portion of the spacecraft's orbit.
- ΔE depends on the relative orientation of the normal nS of the the orbital plane PE of spacecraft with respect to normal nO the orbital plane PO 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 space-craft from the standard orientation in which orbital plane and space-craft's plane are orthogonal. Besides this ΔE depends on the dynamical parameters of the hyperbolic orbit of space-craft given by the conserved energy Etot =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 rS 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 space-craft. 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 θi and θ f between the velocity v of the space-craft with respect to the normal nE of the equatorial plane PE or Earth and the authors are able to give an empirical formula for the energy increment. The angle between PE 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 nO of the orbital plane. For θi ≈ θf the effect is known to be very small. A particular example corresponds to a situation in which one has θi=32 degrees and θf =31 degrees. Obviously the PO≈ PE 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 PE rather than P O. In this case the minimum value of the distance rES between γ E and γS is maximal for the symmetric situation with θi =θf and the effect is minimal. In an asymmetric situation the minimum value of rES 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 PE. 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 non-deformed portion of Earth's orbit:
ΔE/m= V(rS,f)-V(rS,i) ,
Z(rS)= X(γorb;rS)+ X(γd;rS) -X(γnd;rS) ,
X(γi;rS) = ∫γidl/rSi.
Here the subscripts "orb", "d" and "nd" refer to the entire orbit of Earth, to its deformed part, and corresponding non-deformed part. The entire orbit is analogous to a potential of straight string and is expected to give a slowly varying term which is however non-vanishing in the asymmetric situation. The difference of deformed and non-deformed parts gives at large distances dipole type potential behaving like 1/r2 and thus being proportional to v ∞2 by the above expression for the u=rs/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(rS,f)-V(rS,i) ,
V(rS)=-G×(dρ dark/dl)×Z ,
Z(rS)= X(γorb;rS)+ d×cos(Θ)/rS 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 Ed vanishes but End is non-vanishing 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 v02/2 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/r2-factor is also proportional to sin2(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(Θf)-cos(Θi) and this should correspond to the reported behavior. Note that the result vanishes for the symmetric situation in accordance with the empirical findings.
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 BE, 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. Gravito-Lorentz 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 space-craft 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=-vE , where vE is the orbital velocity of Earth. If the tube is thin, the gravito-Lorentz force experienced by dark matter equals in the first approximation to F=-vE × BE with BE evaluated at the axis of the tube. TGD based model for BE (see this) does not allow BE to be a dipole field. BE has only the component Bθ 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 BE at equator is very nearly equal to the strength of the dipole field. The magnitude of BE and thus of F is minimal when the deformation of the tube is in PE, and the deformation occurs very naturally into P E since the non-gravitational forces associated with the dark matter tube must compensate a minimal gravitational force in dynamical equilibrium.
- BθE at equator is in the direction of the spin velocity ω of the Earth. The direction of vE 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 y-axis in the direction of semi-minor and semi-major axes of the Earth's orbit. The corresponding spherical coordinates are defined in an obvious manner. vE is parallel to the tangent vector eφ(t)=-sin (Ωt) ex+ cos(Ωt)ey of the Earth's orbit. The direction of B E at equator is parallel to ω and can be parameterized as eω= cos(θ) ez+ sin(θ)(cos(α) ex+sin(α)ey). F is parallel to the vector -cos(θ) eρ(t) + sin(θ)cos(Ωt-α)ez, where eρ(t) is the unit vector directed from Sun to Earth. The dominant component is directed to Sun.
For TGD based view about astrophysics see the chapter TGD and Astrophysics of "Physics in Many-Sheeted Space-Time".