### A simple quantum model for the formation of astrophysical structures

The stimulus to the latest little calculation in astro-Bohr-orbitology came when Kea mentioned Saturn's rings in her blog. These rings look indeed very quantal and after having recovered from the week-end in the wedding of the son of an old friend, I decided to look for a simple model for them.

Quite generally, the 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 well-understood. 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 particles 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 ruler-and-compass 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 ruler-and-compass 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 space-crafts (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.

- The velocity spectrum for the circular orbits is universal and given as v=v
_{0}/n and that only the radii of Bohr orbits of particles depend on the form of average mass distribution which can vary in wide limits. - Velocity does not decrease with distance and is constant in the presence of only cosmic string.
- The size of the system is always finite and increasing values of n correspond to decreasing radii. This came as a complete surprise, and is a complete opposite for what hydrogen atom like model without cosmic string predicts (when cosmic string is introduced planetary system for given value of v
_{0}has necessarily a finite size).

- First corresponds to a power law, second to a logarithmic velocity distribution and third to a spectrum of orbital radii coming as powers of 2 in accordance with p-adic length scale hypothesis. Fourth mass distribution corresponds to evenly spaced Bohr radii below certain radius.
- Only the third option works as a model for say Earth and predicts that dark matter forms an onionlike structure with the radii of shells coming as powers of 2 (of 2
^{1/2}in the most general formulation of p-adic length scale hypothesis). This prediction is universal and means that dark matter part of stellar objects would be very much analogous to atom having also shell like structure. Actually this is not surprising. - Second and fourth option could define a reasonable model for ring like structures (Saturn's and Jupiter's rings). The predicted universal velocity spectrum for dark rings serves as a test for the model.

P. S. In New Scientist there is an article Do magnetic fields drive dark energy?. Unfortunately, I do not have access to it. Suffice it to say that I have made this identification of dark energy long ago (see for instance this and this), much before the idea about dark matter as large Planck constant phase.

## 2 Comments:

Hi Matti. I am glad you have been thinking about this. But I suspect that people are not going to be convinced by solar system evidence currently available. What about hot jupiter systems? There is a lot of data available now, and I think a few systems with more than 1 planet.

Dear Kea,

I have gone through those extrasolar systems for which one or more planets are known. See TGD and Astrophysics . The fit works with 10 per cent accuracy for all of them. The systems for which the planet is very near to the star correspond to n=1 Bohr orbit (n=3 for Mercury).

The key point is velocity quantization v=v_0/n which is universal in the simple systems considered. This fixes the scale of orbits completely and gravitational Bohr radius GM/v_0 becomes the universal size scale for these systems. This prediction is not possible in GRT or Newtonian theory.

Last night I pondered the basic objection of General Relativist against the model, which is of course is the lack of the manifest General Goordinate and Lorentz invariances. In GRT context this objection would be fatal.

a) In TGD framework one can use Minkowski coordinates of the imbedding space 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 a preferred M^4 coordinates as a rest system(preferred time coordinate) and quantization axis of angular momentum. Fixing quantization axis selects preferred coordinates.

b) One can identify in this system gravitational potential as the g_tt component of metric and define gravi-electric field E_gr uniquely as its gradient.

Consider now the quantization condition.

a) The left hand side of the quantization condition is velocity circulation

Int v*dl .

L is the length L of E^3 projection of the dark matter orbit satisfying geodesic equations of motion (possible problem, perihelion shift of Mercury making orbit slightly non-closed: finite length scale resolution?).

b) Right hand side would be the generalization of

n GM

by the replacement

GM --> Int e*r^2E_gr x dl

e is a unit vector in direction of quantization axis, and r is the radial M^4 coordinate in the preferred system. For Schwartschild metric this reduces to the expected form and reproduces also the contribution of cosmic string correctly.

Everything is Lorentz and General Coordinate Invariant.

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