I finished this paper yesterday. I am exhausted by searching, but finally I von.

The paper can be downloaded from

ResearchGate:

https://www.researchgate.net/publication/321304151_Rotation_Curves_and_Dark_Matter

HAL:

https://hal.archives-ouvertes.fr/hal-01631304

Also it has been announced in viXra:

http://www.vixra.org/abs/1705.0389

Dark Matter is no more.

Dark Matter for beginners:

We need to introduce the Dark Matter (DM) concept because of three reasons:

- To explain the Rotation Curves (RC) of galaxies.
- To explain why the galaxies in clusters are bounded, or at least why they do not escaped from the clusters for cosmological time 10^10 yrs.
- To explain gravitational lensing on clusters of galaxies.

(https://www.scirp.org/journal/PaperInformation.aspx?PaperID=74590) , whereas for the Riemann universe it has value h=9.9x10^-27 , https://arxiv.org/abs/1608.04593 .

As one can see the value of the Planck constant in the Finslerian world coincide with experimentally measured one within the measurement errors of the cosmological parameters (the Hubble constant and cosmological constant are measured up to the second digit).

However we return to the first point. The RCs of the galaxies really look mesmerizing, stick out in the middle of a clean field and I fought for the 8 months until I realized what was happening. First, like everyone, I looked for the RC under the lamp (i.e. rechecked other people’s ideas and looked for errors). I was trying to pull out some new effects from the General Relativity. Than from Finslerian geometry, but anyway final result was very small. I check other ideas, but they did not work. Every time after a couple of months of efforts and calculations, the result did not give the desired effect, remaining extremely small… And then, after eight months, I looked more closely at the Rotation Curves and understood: it’s just a wind! Just a wind tails stretching for the falling matter!

I rushed to make preliminary estimates before getting involved in solving the system of diffusion equations. It turns out the correct value of the effect! Well, and then through the differential equation of diffusion (Fick’s law) showed that Yes, Yes, it does work! Moreover, this agrees perfectly with the recently published experimental results (Phys. Rev. Lett 2016, see also ref. [1] in my paper)!

The problem turned out to be that, the Rotation Curves usually were considered using the simplified scheme, assuming that the motion of all matter is described only by the laws of gravitation. However, in fact, the matter in the galaxies consists of two very different components. The first one is “ideal gas of stars” (collisionless gas that really described by Newton's equations, and is responsible for the inner parts of measured RCs). The second one is real, not ideal gas, made of ions, atoms and molecules. This is the gas that contains neutral hydrogen HI (observed in the 21-cm line), that forms measured in radio outer part of RCs, and that should be described not only by Keplerian law, but also by diffusion differential equations, because it is collisional. This fact can be easily proved. In fact, let the gas concentration (particle in cubic centimeter) be N=10^-3, the thermal velocity V=10^6 cm/s , the scattering cross-section be S=10^-16 cm^2, than the mean free pass time for the atom is t=1/(NVS)=10^13 s = 10^6 yrs. (Here I would like to stress that if the gas is ionized, and this is the case, the cross-section increase and time “t” is reduced by a factor 10 – 100, reinforcing the formation of the wind tail). It is clear that even the time (10^6 yrs) is rather small if compared with characteristic time of the galaxy dynamics, therefore the gas really should be treated as collisional, that is the diffusion equations (Fick’s laws) must be applied.

Therefore, the underlying matter, which rotates approximately in consequence with the Newton's law, and depleted by the star formation process, creates a rarefaction and transmits the angular momentum to the gas of the outer part of disc. This entrains the gas and the gas forms the wind, the wind tail (like that, produced by the metro train in the tunnel) and it is this wind tail, that is considered to be outer part of the rotation curve that “violates” the laws of physics. Thus, the RCs are made from 1) internal part – the rotation of the stellar component governed by Newton's law, and 2) the wind from the external gas, forming “nonphysical tail” of the rotation curve, which was usually explained by the presence of Dark Matter. But now one can see that we do not need the DM concept to explain RC of galaxies.

To find the velocity (RC), we must solve the system of three differential equations (two components of Fick’s first law and one equation expressing the second Fick’s law). I obtained an analytical solution of this system, but it was obtained in form of triple expansion (2D case does not suggests corresponding regular Sturm-Liouville problem, for this reason the 3D case should be solved). This triple expansion, to put it mildly, makes the interpretation to be difficult. But I had to persuade the opponents, so I decided to go another way, namely, to show clearly that the total speed of the gas (Keplerian + diffusion velocity) is related to the observed distribution of gas (column density) through the Fick’s equation. The calculation excellently coincides with the observations (see Fig. 3,4,5,6 in the manuscript).

The general picture is now as follows: there are three regions:

(I) R < R_0, where the stars (collisionless gas) are responsible for dynamics and this part of RC is described by Newton's law;

(II) R_0 < R < R_gas, is a transition region where the active star formation take place, the gas is depleted and many fine details (such as spiral structure, gas temperature, etc. must be taken into account, and

(III): R > R_gas is the region where rotation curves are determined mainly by the gas distribution and Fick’s laws. For this region the rotation speed is V_tot = V_K + V_d, where V_K is the Keplerian velocity and V_d is dynamical velocity of the gas, calculated from the Fick’s laws. In this outer region, the matter is not gravitationally bounded by the host galaxy and moves (is held) also due to the gas pressure (the internal matter, falling and collapsing, pumped out the gas, moving in a spiral downward). It should be noted, that in this outer region, also star formation occurs, and these rare stars are sometimes observed outside of the R_gas and R_25 distances. However, these young stars are doomed to leave the parent galaxy gradually, because they move with the gas velosity V_tot = V_K + V_d, which is larger than Keplerian one.

If you have any questions or comments, I would be very grateful for the discussion below.

09 November 2017