TGD based model explains why the rotation periods of galaxies are same

I learned in FB about very interesting finding about the angular rotation velocities of stars near the edges of the galactic disks (see this). The rotation period is about one giga-year. The discovery was made by a team led by professor Gerhardt Meurer from the UWA node of the International Centre for Radio Astronomy Research (ICRAR). Also a population of older stars was found at the edges besides young stars and interstellar gas. The expectation was that older stars would not be present.

The rotation periods are claimed to in a reasonable accuracy same for all spiral galaxies irrespective of the size. The constant velocity spectrum for distant stars implies ω ∝ 1/r for r>R. It is important do identify the value of the radius R of the edge of the visible part of galaxy precisely. I understood that outside the edge stars are not formed. According to Wikipedia, the size R of Milky Way is in the range (1-1.8)× 10⁵ ly and the velocity of distant stars is v=240 km/s. This gives T∼ R/v∼ .23 Gy, which is by a factor 1/4 smaller than the proposed universal period of T=1 Gy at the edge. It is clear that the value of T is sensitive to the identification of the edge and that one can challenge the identification R_edge=4× R.

In the following I will consider two TGD inspired arguments. The first argument is classical and developed by studying the velocity spectrum of stars for Milky Way, and leads to a rough view about the dynamics of dark matter. Second argument is quantal and introduces the notion of gravitational Planck constant hbar_gr and quantization of angular momentum as multiples of hbar_gr. It allows to predict the value of T and deduce a relationship between the rotation period T and the average surface gravity of the galactic disk.

In the attempts understand how T could be universal in TGD framework, it is best to look at the velocity spectrum of Milky Way depicted in a Wikipedia article about Milky Way (see this).

1. The illustration shows that the v(ρ) has maximum at around r=1 kpc. The maximum corresponds in reasonable approximation to v_max= 250 km/s, which is only 4 per cent above the asymptotic velocity v_rot=240 km/s for distant stars as deduced from the figure.
   

   Can this be an accident? This would suggest that the stars move under the gravitational force of galactic string alone apart from a small contribution from self-gravitation! The dominating force could be due to the straight portions of galactic string determining also the velocity v_rot of distant stars.
   

   It is known that there is also a rigid body part of dark matter having radius r∼ 1 kpc (3.3 × 10³ ly) for Milky Way, constant density, and rotating with a constant angular velocity ω_dark to be identified as the ω_vis at r. The rigid body part could be associated with a separate closed string or correspond to a knot of a long cosmic string giving rise to most of the galactic dark matter.
   

   Remark: The existence of rigid body part is serious problem for dark matter as halo approach and known as core-cusp problem.
   

   For ρ<r stars could correspond to sub-knots of a knotted galactic string and v_rot would correspond to the rotation velocity of dark matter at r when self-gravitation of the knotty structure is neglected. Taking it into account would increase v_rot by 4 per cent to v_max. One would have ω_dark= v_max/r.
   
   

2. The universal rotation period of galaxy, call it T∼ 1 Gy, is assigned with the edge of the galaxy and calculated as T= v(R_edge)/R_edge. The first guess is that the the radius of the edge is R_edge=R, where R∈ (1-1.8)× 10⁵ ly (30-54 kpc) is the radius of the Milky Way. For v(R)= v_rot∼ 240 km/s one has T∼ .225 Gy, which is by a factor 1/4 smaller that T=1 Gy. Taking the estimate T=1 Gy at face value one should have R_edge=4R.
   

   One could understand the slowing down of the rotation if the dark matter above ρ>r corresponds to long - say U-shaped as TGD inspired quantum biology suggests - non-rigid loops emanating from the rigid body part. Non-rigidy would be due to the thickening of the flux tube reducing the contribution of Kähler magnetic energy to the string tension - the volume contribution would be extremely small by the smallness of cosmological constant like parameter multiplying it.
   
   

3. The velocity spectrum of stars for Milky Way is such that the rotation period T_vis=ρ/v_vis(ρ) is quite generally considerably shorter than T=1 Gy. The discrepancy is from 1 to 2 orders of magnitude. The v_vis(ρ) varies by only 17 per cent at most and has two minima (200 km/s and 210 km/s) and eventually approaches v_rot=240 km/s.
   

   The simplest option is that the rotation v(ρ) velocity of dark matter in the range [r,R] is in the first approximation same as that of visible matter and in the first approximation constant. The angular rotation ω would decrease roughly like r/ρ from ω_max to ω_rot=2π/T: for Milky Way this would mean reduction by a factor of order 10^-2. One could understand the slowing down of the rotation if the dark matter above ρ>r corresponds to long - say U-shaped as TGD inspired quantum biology suggests - non-rigid loops emanating from the rigid body part. Non-rigidity would be due to the thickening of the flux tube reducing the contribution of Kähler magnetic energy to the string tension - the volume contribution would be extremely small by the smallness of cosmological constant like parameter multiplying it.
   

   If the stars form sub-knots of the galactic knot, the rotational velocities of dark matter flux loops and visible matter are same. This would explain why the spectrum of velocities is so different from that predicted by Kepler law for visible matter as the illustration of the Wikipedia article shows (see this). Second - less plausible - option is that visible matter corresponds to closed flux loops moving in the gravitational field of cosmic string and its knotty part, and possibly de-reconnected (or "evaporated") from the flux loops.
   

   What about the situation for ρ>R? Are stars sub-knots of galactic knot having loops extending beyond ρ=R. If one assumes that the differentially rotating dark matter loops extend only up to ρ=R, one ends up with a difficulty since v_vis(ρ) must be determined by Kepler's law above ρ=R and would approach v_rot from above rather from below. This problem is circumvented if the loops can extend also to distances longer than R.
   
   

4. Asymptotic constant rotation velocity v_rot for visible matter at r>R is in good approximation proportional to the square root of string tension T_s defining the density per unit length for the dark matter and dark energy of string. v_rot= (2GT_s)^1/2 is determined from Kepler's law in the gravitational field of string. In the article R is identified as the size of galactic disk containing stars and gas.
   
   
5. The universality of T (no dependence on the size R of the galaxy) is guaranteed if the ratio R/r is universal for given string tension T_s. This would correspond to scaling invariance. To my opinion one can however challenge the idea about universality of T since its identification is far from obvious. Rather, the period at r would be universal if the angular velocity ω and perhaps also r are universal in the sense that they depend on the string tension T_s of the galactic string only.
   
   

The above argument is purely classical. One can consider the situation also quantally.

1. The notion of gravitational Planck constant h_gr introduced first by Nottale is central in TGD, where dark matter corresponds to a hierarchy of Planck constants h_eff=n × h. One would have
   

   hbar_gr= GM²/v₀.
   

   for the magnetic flux tubes connecting masses M and m and carrying dark matter. For flux loops from M back to M one would have
   

   hbar_gr= GM²/v₀.
   

   v₀ is a parameter with dimensions of velocity. The first guess is v₀ =v_rot, where v_rot corresponds to the rotation velocity of distant stars - roughly v_rot=4× 10^-3c/5. Distant stars would be associated with the knots of the flux tubes emanating from the rigid body part of dark matter, and T=.25 Gy is obtained for v₀= R/v_rot in the case of Milky Way. The universality of r/R guaranteeing the universality of T would reduce to the universality of v₀.
   
   

2. Assume quantization of dark angular momentum with unit h_gr for the galaxy. Using L = Iω, where I= MR²/2 is moment of inertia, this gives
   

   MR²ω/2= L = m×hbar_gr =2m×GM²/v₀
   

   giving
   

   ω= 2m×hbar_gr/MR² = 2m×GM/(R²v₀)= m× 2πg_gal/v₀ , m=1,2,.. ,
   

   where g_gal= GM/πR² is surface gravity of galactic disk.
   

   If the average surface mass density of the galactic disk and the value of m do not depend on galaxy, one would obtain constant ω as observed (m=1 is the first guess but also other values can be considered).
   
   

3. For the rotation period one obtains
   

   T= v₀/m×g_gal, m=1,2,...
   

   Does the prediction make sense for Milky Way? For M= 10¹²M_Sun represents a lower bound for the mass of Milky Way (see this). The upper bound is roughly by a factor 2 larger. For M=10¹²M_Sun the average surface gravity g_gal of Milky Way would be approximately g_gal ≈ 10^-10g for R= 10⁵ ly and by a factor 1/4 smaller for R= 2× 10⁵ ly. Here g=10 m/s² is the acceleration of gravity at the surface of Earth. m=1 corresponds to the maximal period.
   

   For the upper bound M= 1.5× 10¹²M_Sun of the Milky Way mass (see this) and larger radius R=2× 10⁵ ly one obtains T≈ .23× 10⁹/m years using v₀=v_rot(R/r), R=180r and v_rot=240 km/s.
   
   

4. One can criticize this argument since the rigid body approximation fails. Taking into account the dependence v=v_rotR/ρ in the the integral defining total angular momentum as 2π (M/π R²) ∫ v(ρ) ρ² dρ= Mω R² rather than Mω R²/2 so that the value of ω is reduced by factor 1/2 and the value of T increases by factor 2 to T=.46/m Gy which is rather near to the claimed value of 1 Gy..
   
   

To sum up, the quantization argument combined with the classical argument discussed first allows to relate the value of T to the average surface gravity of the galactic disk and predict reasonably well the value of T.

See the article Four new strange effects associated with galaxies or the chapter TGD and astrophysics.

For a summary of earlier postings see Latest progress in TGD.

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