### Do Bohr rules apply to astrophysical systems?

D. Da Rocha and Laurent Nottale have proposed that Schrödinger equation with Planck constant hbar replaced with what might be called gravitational Planck constant hbar_{gr}= GmM/v_0 (hbar=c=1). v_0 is a velocity parameter having the value v_0=about 145 km/s giving v_0/c=4.6\times 10^{-4}. This is rather near to the peak orbital velocity of stars in galactic halos. Also subharmonics and harmonics of $v_0$ seem to appear. The support for the hypothesis coming from empirical data is impressive. The support for the hypothesis coming from empirical data is impressive. It is surprising that findings of this caliber have not received any attention in popular journals while the latest revolutions in M-theory gain all possible publicity: also I heard from the article by accident from Victor Christianto to whom I am deeply grateful.### Is dark matter in astroscopic quantum state?

Nottale and Da Rocha believe that their Schrödinger equation results from a fractal hydrodynamics. Many-sheeted space-time however suggests that astrophysical systems are not only quantum systems at larger space-time sheets but correspond to a gigantic value of gravitational Planck constant. The gravitational (ordinary) Schrödinger equation would provide a solution of the black hole collapse (IR catastrophe) problem encountered at the classical level. The basic objection is that astrophysical systems are extremely classical whereas TGD predicts macrotemporal quantum coherence in the scale of life time of gravitational bound states. The resolution of the problem inspired by TGD inspired theory of living matter is that it is the dark matter at larger space-time sheets which is quantum coherent in the required time scale. I have proposed already earlier the possibility that Planck constant is quantized and the spectrum is given in terms of logarithms of Beraha numbers B_n= 4cos^2(pi/n): the lowest Beraha number B_3 =1 is completely exceptional in that it predicts infinite value of Planck constant. The inverse of the gravitational Planck constant could correspond a gravitational perturbation of this as 1/hbar_{gr}= v_0/GMm. The general philosophy would be that when the quantum system would become non-perturbative, a phase transition increasing the value of hbar occurs to preserve the perturbative character and at the transition n=4 --> 3 only the small perturbative correction to 1/hbar (3)=0 remains. This would apply to QCD and to atoms with Z>137 as well. TGD predicts correctly the value of the parameter v_0 assuming that cosmic strings and their decay remnants are responsible for the dark matter. The harmonics of v_0 can be understood as corresponding to perturbations replacing cosmic strings with their n-branched coverings so that tension becomes n^2-fold: much like the replacement of a closed orbit with an orbit closing only after n turns. 1/n-sub-harmonic would result when a magnetic flux tube split into n disjoint magnetic flux tubes.### Planetary system as a testing ground

The study of inclinations (tilt angles with respect to the Earth's orbital plane) leads to a concrete model for the quantum evolution of the planetary system. Only a stepwise breaking of the rotational symmetry and angular momentum Bohr rules plus Newton's equation (or geodesic equation) are needed, and gravitational Shrödinger equation holds true only inside flux quanta for the dark matter.- During pre-planetary period dark matter formed a quantum coherent state on the (Z^0) magnetic flux quanta (spherical shells or flux tubes). This made the flux quantum effectively a single rigid body with rotational degrees of freedom corresponding to a sphere or circle (full SO(3) or SO(2) symmetry).
- In the case of spherical shells associated with inner planets the SO(3)--> SO(2) symmetry breaking led to the generation of a flux tube with the inclination determined by m and j and a further symmetry breaking, kind of an astral traffic jam inside the flux tube, generated a planet moving inside flux tube. The semiclassical interpretation of the angular momentum algebra predicts the inclinations of the inner planets. The predicted (real) inclinations are 6 (7) resp. 2.6 (3.4) degrees for Mercury resp. Venus). The predicted (real) inclination of the Earth's spin axis is 24 (23.5) degrees.
- The v_0--> v_0/5 transition necessary to understand the radii of the outer planets can be understood as resulting from the splitting of (Z^0) magnetic flux tube to five flux tubes representing Earth and outer planets except Pluto, whose orbital parameters indeed differ dramatically from those of other planets. The flux tube has a shape of a disk with a hole glued to the Earth's spherical flux shell.
- A remnant of the dark matter is still in a macroscopic quantum state at the flux quanta. It couples to photons as a quantum coherent state but the coupling is extremely small due to the gigantic value of hbar_gr scaling alpha by hbar/hbar_gr: hence the darkness. Note however that it is the entire condensate that couples to electromagnetism with this coupling, individual charged particles couple normally.

### Living matter and dark matter

The most interesting predictions from the point of view of living matter are following.- The dark matter is still there and forms quantum coherent structures of astrophysical size. In particular, the (Z^0) magnetic flux tubes associated with the planetary orbits define this kind of structures. The enormous value of h_{gr} makes the characteristic time scales of these quantum coherent states extremely long and implies macro-temporal quantum coherence in human and even longer time scales.
- The rather amazing coincidences between basic bio-rhythms and the periods associated with the states of orbits in solar system suggest that the frequencies defined by the energy levels of the gravitational Schrödinger equation might entrain with various biological frequencies such as the cyclotron frequencies associated with the magnetic flux tubes. For instance, the period associated with n=1 orbit in the case of Sun is 24 hours within experimental accuracy for v_0: the duration of day in Earth and in a good approximation also in Mars! Second example is the mysterious 5 second time scale associated with the Comorosan effect.