There are two objections against planetary Bohr orbitology.
In the previous posting I proposed a simple model explaining why inner and outer planets must have different values of v0 by taking into account cosmic string contribution to the gravitational potential which is negligible nowadays but was not so in primordial times. Among other things this implies that planetary system has a finite size, at least about 1 ly in case of Sun (nearest star is at distance of 4 light years).
I have also applied the quantization rules to exoplanets in the case that the central mass and orbital radius are known. Errors are around 10 per cent for the most favored value of v0=2-11 (see this). The "anomalous" planets with very small orbital radius correspond to n=1 Bohr orbit (n=3 is the lowest orbit in solar system). The universal velocity spectrum v= v0/n in simple systems perhaps the most remarkable prediction and certainly testable: this alone implies that the Bohr radius GM/v02 defines the universal size scale for systems involving central mass. Obviously this is something new and highly non-trivial.
The recently observed dark ring in MLy scale is a further success and also the rings and Moons of Saturn and Jupiter obey the same universal length scale (n≥ 5 and v0→ (16/15)×v0 and v0→ 2×v0).
There is a further objection. For our own Moon orbital radius is much larger than Bohr radius for v0=2-11: one would have n≈138. n≈7 results for v0 →v0/20 giving r0≈ 1.2 RE. The small value of v0 could be understood to result from a sequence of phase transitions reducing the value of v0 to guarantee that solar system participates in the average sense to the cosmic expansion and from the fact inner planets are older than outer ones in the proposed scenario.
Remark: Bohr orbits cannot participate in the expansion which manifests itself as the observed apparent shrinking of the planetary orbits when distances are expressed in terms Robertson-Walker radial coordinate r=rM. This anomaly was discovered by Masreliez and is discussed here. Ruler-and-compass hypothesis suggests preferred values of cosmic times for the occurrence of these transitions. Without this hypothesis the phase transitions could form almost continuum.
2. How General Coordinate Invariance and Lorentz invariance are achieved?
One can use Minkowski coordinates of the M4 factor of the imbedding space H=M4×CP2 as preferred space-time coordinates. The basic aspect of dark matter hierarchy is that it realizes quantum classical correspondence at space-time level by fixing preferred M4 coordinates as a rest system. This guarantees preferred time coordinate and quantization axis of angular momentum. The physical process of fixing quantization axes thus selects preferred coordinates and affects the system itself at the level of space-time, imbedding space, and configuration space (world of classical worlds). This is definitely something totally new aspect of observer-system interaction.
One can identify in this system gravitational potential Φgr as the gtt component of metric and define gravi-electric field Egr uniquely as its gradient. Also gravi-magnetic vector potential Agr and and gravimagnetic field Bgrcan be identified uniquely.
3. Quantization condition for simple systems
Consider now the quantization condition for angular momentum with Planck constant replaced by gravitational Planck constant hbargr= GMm/v0 in the simple case of pointlike central mass. The condition is
m∫ v•dl = n × hbargr.
The condition reduces to the condition on velocity circulation
∫ v•dl = n × GM/v0.
In simple systems with circular rings forced by Zn symmetry the condition reduces to a universal velocity spectrum
so that only the radii of orbits depend on mass distribution. For systems for which cosmic string dominates only n=1 is possible. This is the case in the case of stars in galactic halo if primordial cosmic string going through the center of galaxy in direction of jet dominates the gravitational potential. The velocity of distant stars is correctly predicted.
Zn symmetry seems to imply that only circular orbits need to be considered and there is no need to apply the condition for other canonical momenta (radial canonical momentum in Kepler problem). The nearly circular orbits of visible matter objects would be naturally associated with dark matter rings or more complex structures with Zn symmetry and dark matter rings could suffer partial or complete phase transition to visible matter.
4. Generalization of the quantization condition