### Dark matter based model for Pioneer and flyby anomalies

This has been very enjoyable period for dark matter afficionado. During last month I have had an opportunity to apply TGD based vision about dark matter to about five existing or completely new anomalies. Just yesterday I learned about the new findings related to Pioneer and flyby anomalies which challenge the standard theory of gravitation.

I have proposed earlier a model for Pioneer anomaly resulting as a by-product of an explanation of another anomaly which can be understood if cosmic expansion is compensated by a radial contraction of solar system in local Robertson-Walker coordinates. The recent findings reported here allow to sharpen the model suggesting a universal primordial mass density associated with the solar system. The facts about flyby anomaly lead to a model in which anomalous energy gain of the space-craft in Earth-craft rest system results when it passes through a spherical layer of dark matter containing Earth's orbit (this is of course too stringent model). These structures are predicted by the model explaining the Bohr quantization of planetary orbits to served as templates for the condensation of visible matter around them.

** 1. Explanation of Pioneer anomaly in terms of dark matter**

I have proposed an explanation of Pioneer anomaly as a prediction of a model explaining the claimed radial acceleration of planets which is such that it compensates the cosmological expansion of planetary system. The correct prediction is that the anomalous acceleration is given by Hubble constant. The precise mechanism allowing to achieve this was not proposed.

A possible mechanism is based the presence of dark matter increasing the effective solar mass. Since acceleration anomaly is constant, a dark matter density behaving like ρ_{d} = (3/4π)(H/Gr), where H is Hubble constant giving M(r) propto r^{2}, is required. For instance, at the radius R_{J} of Jupiter the dark mass would be about (δa/a) M_{S}≈ 1.3× 10^{-4}M_{S} and would become comparable to M_{S} at about 100R_{J}=520 AU. Note that the standard theory for the formation of planetary system assumes a solar nebula of radius of order 100AU having 2-3 solar masses. For Pluto at distance of 38 AU the dark mass would be about one per cent of solar mass. This model would suggest that planetary systems are formed around dark matter system with a universal mass density. For this option dark matter could perhaps be seen as taking care of the contraction compensating for the cosmic expansion by using a suitable dark matter distribution.

Here the possibility that the acceleration anomaly for Pioneer 10 (11) emerged only after the encounter with Jupiter (Saturn) is raised. The model explaining Hubble constant as being due to a radial contraction compensating cosmic expansion would predict that the anomalous acceleration should be observed everywhere, not only outside Saturn. The model in which universal dark matter density produces the same effect would allow the required dark matter density ρ_{d}= (3/4π)(H/Gr) be present only as a primordial density able to compensate the cosmic expansion. The formation of dark matter structures could have modified this primordial density and visible matter would have condensed around these structures so that only the region outside Jupiter would contain this density.
There are also diurnal and annual variations in the acceleration anomaly (see the same article). These variations should be due to the physics of Earth-Sun system. I do not know whether they can be understood in terms of a temporal variation of the Doppler shift due to the spinning and orbital motion of Earth with respect to Sun.

** 2. Explanation of Flyby anomaly in terms of dark matter**

The so called flyby anomaly might relate to the Pioneer 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 **e**_{z} · **r**× **v**.

Here e is total energy per mass for the space-craft, 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 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 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**·δ**v** 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 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.

For instance, if the space-craft traverses shell structure, its kinetic energy per mass in Earth cm system changes by a constant amount not depending on the mass of the space-craft:

δE/m ≈ v_{inf,E}×δv= δV_{gr} = GδM_{eff,S}/R.

Here R is the outer radius of the shell and v_{inf,E} is the magnitude of asymptotic velocity in Earth cm system. This very simple prediction should be testable. If the space-craft arrives from the direction of Sun the energy increases. If the space-craft returns back to the sunny side, the net anomalous energy gain vanishes. This has been observed in the case of Pioneer 11 encounter with Jupiter (see this).

The mechanism would make it possible to deduce the total dark mass of, say, spherical shell of dark matter. One has

δM/M_{S} ≈δv/v_{inf,E} ×(2K/V) , K= v^{2}_{inf,E}/2 , V=GM_{S}/R .

For the case considered δM/M_{S}≥ 2× 10^{-6} is obtained. Note that the amount of dark mass within sphere of 1 AU implied by the explanation of Pioneer anomaly would be about 6.2× 10^{-6}M_{S} from Pioneer anomaly. Since the orders of magnitude are same one might consider the possibility that the primordial dark matter has concentrated in spherical shells in the case of inner planets as indeed suggested by the model for quantization of radii of planetary orbits.

In the solar cm system the energy gain is not constant. Denote by **v**_{i,E} and **v**_{f,E} the initial and final velocities of the space-craft in Earth cm. Let δ**v** be the anomalous change of velocity in the encounter and denote by θ the angle between the asymptotic final velocity **v**_{f,S} of planet in solar cm. One obtains for the corrected e_{g,S} the expression

e_{g,S}= (1/2)[(**v**_{f,E}+**v**_{P}+δ**v**)^{2}-(**v**_{i,E}+**v**_{P})^{2}] .

This gives for the change δe_{g,S}

δe_{g,S}≈(**v**_{f,E}+**v**_{P})· δ**v**≈ v_{f,S} δv× cos(θ_{S})= v_{inf,S}δv × cos(θ_{S}).

Here v_{inf,S} is the asymptotic velocity in solar cm system and in excellent approximation predicted by the theory.
Using spherical shell as a model for dark matter one can write this as

δe_{g,S}= v_{inf,S}/v_{inf,E} × G δM/R × cos(θ_{S}) .

The proportionality of δe_{g,S} to cos(θ_{S}) should explain the variation of the anomalous energy gain.

For a spherical shell δ**v** is in the first approximation orthogonal to v_{P} since it is produced by a radial acceleration so that one has in good approximation

δe_{g,S}≈**v**_{f,S}· δ**v**≈ **v**_{f,E}· δ**v**≈ v_{f,S}δv × cos(θ_{S})= v_{inf,E} δv× cos(θ_{E}).

For Cassini and Messenger cos(θ_{S}) should be rather near to zero so that v_{inf,E} and v_{inf,S} should be nearly orthogonal to the radial vector from Sun in these cases. This provides a clear cut qualitative test for the spherical shell model.

For TGD based view about astrophyscs see the chapter TGD and Astrophysics of "Physics in Many-Sheeted Space-Time".

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