The dark matter ring found by NASA corresponds to the lowest Bohr orbit

I told yesterday about dark matter ring found by NASA. TGD based model for dark matter inspires the hypothesis that it corresponds to Bohr orbit for macroscopically quantum coherent dark matter with gigantic value of Planck constant predicted by the model. The article about finding is now in archive and contains the data making possible to test the model. I am grateful for Kea for providing the link. The ring corresponds with a good accuracy to the lowest Bohr orbit for v₀= 3×2^-11, which is 3 times the favored value but allowed by the general hypothesis for the favored values of Planck constant.

I added the first version of the little calculation to the previous posting. Unfortunately it contained besides an innocent error in the formula of Bohr radius also a numerical error giving a result which was exactly 10 times too small. The erratic calculation however happened to give the correct result for v₀=2^-11, which is the preferred value. In some magic manner mistakes conspire to give the desired result and ridicule the poor theoretician! To minimize confusion I deleted the original calculation and added the corrected calculation here.

The number theoretic hypothesis for the preferred values of Planck constants states that the gravitational Planck constant

hbar= GMm/v₀

equals to a ruler-and-compass rational which is ratio q= n₁/n₂ of ruler-and-compass n_i integers expressible as a product of form n=2^k∏ F_s, where all Fermat primes F_s are different. Only four of them are known and they are given by 3, 5, 17, 257, 2¹⁶+1. v₀=2^-11 applies to inner planets and v₀=2^-11/5 to outer planets and the conditions from the quantization of hbar are satisfied.

The obvious TGD inspired hypothesis is that the dark matter ring corresponds to Bohr orbit. Hence the distance would be

r= n² r₀,

where r₀ is Bohr radius and n is integer. n=1 for lowest Bohr orbit. The Bohr radius is given

r₀=GM/v₀²,

where M the total mass in the dense core region inside the ring. This would give distance of about 2000 times Schwartschild radius for the lowest orbit for the preferred value of v₀=2^-11.

This prediction can be confronted with the data since the article Discovery of a ringlike dark matter structure in the core of the galaxy cluster C1 0024+17 is in the archive now.

1. From the Summary and Conclusion part of the article the radius of the ring is about .4 Mpc, which makes in a good approximation 1.2 Mly (I prefer light years). More precisely - using arc second as a unit - the ring corresponds to a bump in the interval 60''-85'' centered at 75''. Figure 10 of of the article gives a good idea about the shape of the bump.

2. From the article the mass in the dense core within radius which is almost half of the ring radius is about M=1.5×10¹⁴× M_Sun. The mass estimate based on gravitational lensing gives M=1.5×10¹⁴× M_Sun. If the gravitational lensing involves dark mass not in the central core, the first value can be used as the estimate. The Bohr radius this system is r₀=GM/v₀²= 1.5×10¹⁴× r₀(Sun),

   where I have assumed v₀=2^-11 as for the inner planets in the model for the solar system.

3. The Bohr orbit for our planetary system predicts correctly Mercury's orbital radius as n=3 Bohr orbit for v₀ =2^-11 so that one has

   r₀(Sun)=r_M/9,

   where r_M is Mercury's orbital radius. One obtains

   r₀= 1.5×10¹⁴× r_M/9.

4. Mercury's orbital radius is in a good approximation r_M=.4 AU, and AU (the distance of Earth from Sun) is 1.5×10¹¹ meters. 1 ly corresponds to .95×10¹⁶ meters. This gives

   r₀ =11 Mly to be compared with 1.2 Mly deduced from observations. The result is by a factor 9 too large.

5. If one replaces v₀ with 3v₀ one obtains downwards scaling by a factor of 1/9, which gives r₀=1.2 Mly. The general hypothesis indeed allows to scale v₀ by a factor 3.

6. If one considers instead of Bohr orbits genuine solutions of Schrödinger equation then only n> 1 structures can correspond to rings like structures. Minimal option would be n=2 with v₀ replaced with 6v₀.

The conclusion would be that the ring would correspond to the lowest possible Bohr orbit for v₀=3× 2^-11. I would have been really happy if the favored value of v₀ had appeared in the formula but the consistency with the ruler-and-compass hypothesis serves as a consolation. Skeptic can of course always argue that this is a pure accident. If so, it would be an addition to long series of accidents (planetary radii in solar system and radii of exoplanets). One can of course search rings at radii corresponding to n=2,3,... If these are found, I would say that the situation is settled.

For more details see the new chapter Quantum Astrophysics of "Classical Physics in Many-Sheeted Space-time"