Pioneer and Flyby anomalies as demonstrations of dark matter spheres associated with orbits of planets

There was a very interesting link at Thinking Allowed Original in this morning - a lot of thanks for Ulla. The link was to two old anomalies discovered in the solar system: Pioneer anomaly and Flyby anomalies with which I worked for years ago. I remember only the general idea that dark matter concentrations at orbits of planets or at spheres with radii equal that of orbit could cause the anomaly. So I try to reconstruct all from scratch and during reconstruction become aware of something new and elegant that I could not discover for years ago.

The article says that Pioneer anomaly is understood. I am not at all convinced about the solution of Pioneer anomaly. Several "no new physics" solutions have been tailored during years but later it has been found that they do not work.

Suppose that dark matter is at the surface of sphere so that by a well-known text book theorem it does not create gravitational force inside it. This is an overall important fact, which I did not use earlier. The model explains both anomalies and also allow to calculate the total amount of dark matter at the sphere.

1. Consider first Pioneer anomaly.
   
   
   1. Inside the dark matter sphere with radius of Jupiter's orbit the gravitational force caused by dark matter vanishes. Outside the sphere also dark matter contributes to the gravitational attraction and Pioneer's acceleration becomes a little bit smaller since the dark matter at the sphere containing the orbit radius of Jupiter or Saturn also attracts the space-craft after the passby. A simple test for spherical model is the prediction that the mass of Jupiter effectively increases by the amount of dark matter at the sphere after passby.
      
      
   2. The magnitude of the Pioneer anomaly is about Δa/a=1.3× 10^-4 and translates to M_dark/M≈ 1.3× 10^-4. What is highly non-trivial is that the anomalous acceleration is given by Hubble constant suggesting that there is a connection with cosmology fixing the value of dark mass once the area of the sphere containing it is fixed. This follows as a prediction if the surface mass density is universal and proportional to the Hubble constant. The value of acceleration is a=.8× 10^-10× g, g=9.81 m/s² whereas the MOND model () finds optimal value for the postulated minimal gravitational acceleration to be a₀=1.2× 10^-10 m/s². In TGD framework it would be assignable to the traversal through a dark matter shell. The ratio of the two accelerations is a/a₀=6.54.
      

      Could one interpret the equality of the two accelerations as an equilibrium condition? The Hubble acceleration associated with the cosmic expansion (expansion velocity increases with distance) would be compensated by the acceleration due to the gravitational force of dark matter? Could one interpret the equality of the two accelerations as an equilibrium condition? The Hubble acceleration H associated with the cosmic expansion (expansion velocity increases with distance) would be compensated by the acceleration due to the gravitational force of dark matter. The formula for surface density of dark matter is from Newton's law GM_dark/R₂= H given by σ_dark= H/4π G. The approximate value of dark matter surface density is from Hc=6.7 × 10^-10 m/s² equal to σ =.8 kg/m² and surprisingly large.
      
      
      

   3. TGD inspired quantum biology requiring that the universal cyclotron energy spectrum of dark photons h_eff=h_gr transforming to bio-photons is in visible and UV range for charged particles gives the estimate M_dark/M_E≈ 2× 10^-4 and is of the same order of magnitude smaller than for Jupiter. The minimum value of the magnetic field at flux tubes has been assumed to be B_E=.2 Gauss, which is the value of endogenous magnetic field explaining the effects of ELF em radiation on vertebrate brain. The two estimates are clearly consistent.
      
      
2. In Flyby anomaly spacecraft goes past Earth to gain momentum (Earth acts as a sling) for its travel towards Jupiter. During flyby a sudden acceleration occurs but this force is on only during the flyby but not before or after that. The basic point is that the spacecraft visits near Earth, and this is enough to explain the anomaly.
   

   The space-craft enters from a region outside the orbit of Earth containing dark matter and thus experiences also the dark force created by the sphere. After that the space craft enters inside the dark matter region, and sees a weaker gravitational force since the dark matter sphere is outside it and does not contribute. This causes a change in its velocity. After flyby the spacecraft experiences the forces caused by both Earth and dark matter sphere and the situation is the same as before flyby. The net effect is a change in the velocity as observed. From this the total amount of dark matter can be estimated. Also biology based argument gives an estimate for the fraction of dark matter in Earth.
   

This model supports the option in which the dark matter is concentrated on sphere. The other option is that it is concentrated at flux tube around orbit: quantitative calculations would be required to see whether this option can work. One can consider of course also more complex distributions: say 1/r distribution giving rise to constant change in acceleration outside the sphere.

A simple TGD model for the sphere containing dark matter could be in terms of boundary defined by gigantic wormhole contact (at its space-time sheet representing "line of generalized Feynman diagram" one has deformation of CP₂ type vacuum extremal with Euclidian signature of induced metric) with radius given by the radius of Bohr orbit with gravitational Planck constant equal to h_gr =GMm/v₀. This radius does not depend on the mass of the particle involved and is given by r_n= GM/v₀³ where 2GM is Schwartschild radius equal to 3 km for Sun. One has v₀/c≈ 2^-11.

An interesting possibility is that also Earth-Moon system contains a spherical shell of dark matter at distance given by the radius of Moon's orbit (about 60 Earth's radii). If so the analogs of the two effects could be observed also in Earth Moon system and the testing of the effects would become much easier. This would also mean understanding of the formation of Moon. Also interior of Earth (and also Sun) could contain spherical shells containing dark matter as the TGD inspired model for the spherically symmetric orbit constructed for more than two decades ago suggests. One can raise interesting questions. Could also the matter in mass scale systems be accompanied by dark matter shells at radii equal to Bohr radii in the first approximation and could these effects be tested? Note that a universal surface density for dark matter predicts that the change of acceleration universally be given by Hubble constant H.

For background and details see the chapter TGD and Astrophysics of "Physics in Many-Sheeted Space-time" or the article Pioneer and Flyby anomalies for almost ten years later.