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)× 10^{5} 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).

- 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 × 10

^{3}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.

- 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)× 10^{5}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.

- 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.

- 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.

- 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 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}= GM^{2}/v_{0}.

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}= GM^{2}/v_{0}.

v

_{0}is a parameter with dimensions of velocity. The first guess is v_{0}=v_{rot}, where v_{rot}corresponds to the rotation velocity of distant stars - roughly v_{rot}=4× 10^{-3}c/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 v_{0}= 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 v_{0}.

- Assume quantization of dark angular momentum with unit h
_{gr}for the galaxy. Using L = Iω, where I= MR^{2}/2 is moment of inertia, this gives

MR

^{2}ω/2= L = m×hbar_{gr}=2m×GM^{2}/v_{0}

giving

ω= 2m×hbar

_{gr}/MR^{2}= 2m×GM/(R^{2}v_{0})= m× 2πg_{gal}/v_{0}, m=1,2,.. ,

where g

_{gal}= GM/πR^{2}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).

- For the rotation period one obtains

T= v

_{0}/m×g_{gal}, m=1,2,...

Does the prediction make sense for Milky Way? For M= 10

^{12}M_{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=10^{12}M_{Sun}the average surface gravity g_{gal}of Milky Way would be approximately g_{gal}≈ 10^{-10}g for R= 10^{5}ly and by a factor 1/4 smaller for R= 2× 10^{5}ly. Here g=10 m/s^{2}is the acceleration of gravity at the surface of Earth. m=1 corresponds to the maximal period.

For the upper bound M= 1.5× 10

^{12}M_{Sun}of the Milky Way mass (see this) and larger radius R=2× 10^{5}ly one obtains T≈ .23× 10^{9}/m years using v_{0}=v_{rot}(R/r), R=180r and v_{rot}=240 km/s.

- One can criticize this argument since the rigid body approximation fails. Taking into account the dependence v=v
_{rot}R/ρ in the the integral defining total angular momentum as 2π (M/π R^{2}) ∫ v(ρ) ρ^{2}dρ= Mω R^{2}rather than Mω R^{2}/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..

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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