A simple quantum model for the formation of astrophysical structuresThe mechanisms behind the formation of planetary systems, galaxies and larger systems are poorly understood but planar structures seem to define a common denominator and the recent discovery of dark matter ring in a galactic cluster in Mly scale (see this) suggest that dark matter rings might define a universal step in the formation of astrophysical structures. Also the dynamics in planet scale is poorly understood. In particular, the rings of Saturn and Jupiter are very intricate structures and far from wellunderstood. Assuming spherical symmetry it is far from obvious why the matter ends up to form thin rings in a preferred plane. The latest surprise is that Saturn's largest, most compact ring consist of clumps of matter separated by almost empty gaps. The clumps are continually colliding with each other, highly organized, and heavier than thought previously. The situation suggests that some very important piece might be missing from the existing models, and the vision about dark matter as a quantum phase with a gigantic Planck constant (see this and this) is an excellent candidate for this piece. The vision that the quantum dynamics for dark matter is behind the formation of the visible structures suggests that the formation of the astrophysical structures could be understood as a consequence of Bohr rules. 1. General quantum vision about formation of structures The basic observation is that in the case of a straight cosmic string creating a gravitational potential of form v_{1}^{2}/ρ Bohr quantization does not pose any conditions on the radii of the circular orbits so that a continuous mass distribution is possible. This situation is obviously exceptional. If one however accepts the TGD based vision (see this) that the very early cosmology was cosmic string dominated and that elementary p"/public_html/articles/ were generated in the decay of cosmic strings, this situation might have prevailed at very early times. These cosmic strings can transform to strings with smaller string tension and magnetic flux tubes can be seen as their remnants dark energy being identifiable as magnetic energy. If so, the differentiation of a continuous density of ordinary matter to form the observed astrophysical structures would correspond to an approach to a stationary situation governed by Bohr rules and in the first approximation one could neglect the intermediate stages. Cosmic string need not be infinitely long: it could branch into n return flux tubes, n very large in accordance with the Z_{n} symmetry for the dark matter but also in this case the situation in the nearby region remains the same. For large distances the whole structure would behave as a single mass point creating ordinary Newtonian gravitational potential. Also phase transitions in which the system emits magnetic flux tubes so that the contribution of the cosmic string to the gravitational force is reduced, are possible. What is of utmost importance is that the cosmic string induces the breaking of the rotational symmetry down to a discrete Z_{n} symmetry and in the presence of the central mass selects a unique preferred orbital plane in which gravitational acceleration is parallel to the plane. This is just what is observed in astrophysical systems and not easily explained in the Newtonian picture. In TGD framework this relates directly to the choice of quantization axis of angular momentum at the level of dark matter. This mechanism could be behind the formation of planar systems in all length scales including planets and their moons, planetary systems, galaxies, galaxy clusters in the scale of Mly, and even the concentration of matter at the walls of large voids in the scale of 100 Mly. The Z_{n} symmetry for the dark matter with very large n suggests the possibility of more precise predictions. If n is a rulerandcompass integer it has as factors only first powers of Fermat primes and a very large power of 2. The breaking of Z_{n} symmetry at the level of visible matter would naturally occur to subgroups Z_{m} subset Z_{n}. Since m is a factor of n, the average number of matter clumps could tend to be a factor of n, and hence a rulerandcompass integer. Also the hexagonal symmetry discovered near North Pole of Saturn (see this)could have interpretation in terms of this symmetry breaking mechanism. 2. How inner and outer planets might have emerged? The Bohr orbit model requires different values of the parameter v_{0} related by a scaling v_{0}→v_{0}/5 for inner and outer planets. It would be nice to understand why this is the case. The presence of cosmic string along rotational axis implied both by the model for the asymptotic state of the star and TGD based model for gamma ray bursts might allow to understand this result. One can construct a simple modification of the hydrogen atom type model for solar system by including the contribution of cosmic string to the gravitational force. For circular orbits the condition identifying kinetic and gravitational radial accelerations plus quantization of angular momentum in units of gravitational Planck constant are used. The prediction is that only a finite number of Bohr orbits are possible. One might hope that this could explain the decomposition of the planetary system to inner and outer planets. String tension implies anomalous acceleration of same form as the radial kinetic acceleration implying that for given radius kinetic energy per mass is shifted upwards by a constant amount. This acceleration anomaly is severely bounded above by the constant acceleration anomaly of spacecrafts (Pioneer anomaly) and for the recent value of the cosmic string tension the number of allowed inner planets is much larger than 3. The situation was however different in the primordial stage when cosmic string tension was much larger and gradually reduced in phase transitions involving the emission of closed magnetic flux tubes. Primordial Sun could have emitted the seeds of the two planetary systems related by scaling and that this might have happened in the phase transition reducing magnetic flux by the emission of closed magnetic flux tube structure. 3. Models for the interior of astrophysical object and for planetary rings Using similar quantization conditions one can construct a very simple model of astrophysical object as a cylindrically symmetric pancake like structure. There are three basic predictions which do not depend on the details of the mass distribution.
