Dedicated to the Dark Matter Day.

In this text it is argued that there is no need for Dark Matter (DM) in astrophysics and cosmology. All subsequent information is based on the article “** Gas kinetics of galactic disks explains rotation curves of S-type galaxies without a need for dark matter**”, published in the International Journal of Modern Physics A (2022) v.37, 2250171.

It is well known that DM is required mainly for solving three problems. These are: 1) “non -physical” Rotation Curves (RC) of galaxies, 2) Observed galaxy clusters, indicating that galaxies are gravitationally bounded and belong to their cluster at least for a time comparable to the cosmological one, and 3) Observed gravitational lensing on galaxy clusters.

The first item of this list represents the greatest difficulty, since it clearly shows us that we miss something essential at the level of the fundamentals of physics. The problems mentioned in items 2 and 3 are not so crucial and they can be solved without the involvement of DM. They find their solution within the framework of standard physics, since there is a fairly large space for the maneuver (for details, see the introduction of the ** article under discussion**).

Another additional item should also be mentioned here. In accordance with the point of view of cosmologists, DM must exist precisely so that the observed structures of the Universe could be formed during cosmological time within the framework of Riemannian geometry. However, on the one hand, the geometry of Riemann is a very specific case of a more general Finsler geometry. On the other hand, no one promised us that we would be born precisely in the universe described by Riemannian geometry. Therefore, such an attempt to preserve the right to work precisely in Riemann geometry (by postulating the existence of DM) looks a few strange. I reasonably believe that the geometry of our Universe is not Riemannian one. In this case, many cosmological problems that arise in the Riemannian world simply disappear if models are built for the universe described by a more general geometry with asymmetric connections (in the general case the Finslerian geometry). For this reason, within the framework of Finslerian geometry (where cosmological time is much larger than that in the Universe, described by the geometry of Riemann), there is enough time to form observable structures in the universe and therefore there is no need to involve the DM concept. The problem of the real geometry of our Universe I will consider next time, and today I analyze the first point of the list – the problem of the rotation curves of spiral galaxies.

It is well known that until the middle of the 20th century, the observation of galaxies was carried out only through optical telescopes. With the development of radioastronomy in the second half of the 20th century, it became possible to observe galactic gas in the molecular lines and in line of neutral hydrogen (21 cm). This made it possible to draw up detailed maps of ** “non-physical” rotation curves** for the outer parts of galactic disks observed in radio. Sometimes the rotation curves (RC) extend to hundreds of kiloparsecs (one parsec = 3 light years). In Fig. 1, one can see a typical rotation curve of a spiral galaxy.

Fig. 1. Typical observed rotation curve of spiral galaxy (green open circles) and Keplerian one (red solid line). One can see enormous discrepancy between observed and Keplerian RCs at large distances (outer part of galactic disk).

Of course, there are also optical observations of rare and remote stars which populate the outer regions, but it is not difficult to show that they will be bounded by host galaxy and will move along elliptical Keplerian orbits. Therefore, their movements are easily explained within the framework of classical mechanics. I will return to this topic a little later.

Consider a typical spiral galaxy. Since the purpose of this work is to explain the rotation curves of galaxies without involving the concept of DM, then below I will mean by the word “matter” only baryonic matter, reasonably believing that the DM does not exist (because it will not be required to explain the rotation curves of galaxies).

It is well known that the matter in galaxies consists mainly of stars (located predominantly in the visible part of host galaxy) and neutral hydrogen (HI) that dominates the outer parts of the disks of galaxies, in whose lines the RCs for the outer parts of galaxies are observed at 21 cm.

It is rightly believed that the stellar component can be described as a collisionless gas consisting of ideal particles moving in a gravitational field. Until now, it has also been generally accepted that when describing rotation curves, neutral hydrogen can also be described as a collisionless gas interacting only gravitationally. Thus, the problem of widely used models is that the dynamics of galactic disks is calculated according to an overly simplified scheme, assuming that the global movement of all matter in the galactic disk (we speak about RC) is described only by the laws of Newtonian gravity, unreasonably believing that gas kinetics does not affect the formation of rotation curves. It should be recognized that attempts were made to estimate the contribution of gas kinetics to global dynamics, but this was erroneously done based on the equations of hydrodynamics (HD), which cannot be applied to describe a rarefied gas. Also, all relations following from the HD equations and used for a simplified description of a rarefied gas become inapplicable. Since such HD simulations were done outside the scope of the HD equations applicability, they cannot be considered as a credible argument.

To understand where the mistake was made, let’s see how the HD equations are derived. I follow here the standard derivation of the HD equations suggested in the textbook (volume X of the theoretical physics Lifshitz, Pitaevsky “Physical Kinetics” section 5).

The kinetic equation for one kind of particles is:

To derive the continuity equation, one should integrate the kinetic equation over the momenta of the particles. However, integration over momenta can only be performed under the assumption that all parameters system (such as density, velocities, temperature, etc.) change little over distances of the order of the particle’s characteristic mean free path *l _{fp}*. In other words, from the very beginning, the domain of applicability of the HD equations is restricted by the case in which the characteristic length for the system is L

_{k}>>

*l*. Only in this case the continuity equation can be derived and will make sense:

_{fp}To derive the rest of the HD equations, we should multiply the kinetic equation (1) by the momentum (or energy) and then integrate over the momentum. In this case, to perform the integration, one must set the same restrictions (3):

For this reason (and given that we are only interested in the first equation, since in the case of rotation curves we are dealing with a steady, quasi-stationary motion), we can omit the consideration of equations with higher derivatives. There is nothing new or useful for us there.

Thus, the most important point here is that the HD equations can be derived (and therefore can be applied) if and only if the condition (3) is satisfied, which imposes a serious limitation on the applicability of the HD equations.

In the case of calculating the dynamics of gas in the outer parts of galactic disk, we are faced with the kinetics and dynamics of a very rarefied gas, for which condition (3) is no longer satisfied. It means that the HD equations are not applicable (their use leads to an unphysical result). If the HD equations themselves are not applicable, then what should be done? There are many options, I will point out only two obvious ones:

1) Apply directly more general kinetic equations. (this is a cumbersome option).

2) By analogy with the derivation of HD equations, derive more general equations that would be devoid of the indicated disadvantages (simple and clear way). In the work under discussion, I follow this path.

The task of finding the rotation curves of galaxies will be significantly simplified if we consider the fact that a spiral galaxy is a pseudo-stationary structure. In other words, the disk of the galaxy demonstrates a steady state solution. In the case described by the HD equations, this corresponds to a steady flow around an object. It is well known that in this case there is no need to solve all the HD equations. Such steady flows are described by a single continuity equation. Thus, in the case of a steady flow of rarefied gas in a galaxy, the solution is also will be given by one equation (some analogue of the continuity equation). Let me briefly outline the derivation of the equation. Let’s start with kinetic equations as we did in the case of the HD approach.

To obtain equation (2), we assumed condition (3) be satisfied everywhere. However, this is not the case for extremely rarefied gas. The quantity (*v ^{α}f*) changes significantly over the mean free path. For this reason, the kinetic equation cannot be averaged over the momenta. Beside this, we are interested in the rotation curve, which represents a global movement of gas. Moreover, we are not at all interested in small-scale turbulent motions. For these reasons, one should average equations (1) over the volume, excluding small-scale motions and making the equations insensitive to restrictions (3):

First term gives *dm/dt* , where m is mass of gas inside of the integrating volume. Second term can be transformed to

Finally we obtain:

This is a well-known diffusion equation, which in our case serves as a replacement for the continuity equation. The equation (5) no longer suffers from the constraint (3) inherent in the HD approach, and therefore it can be applied to calculate large-scale movements of very rarefied gas (rotation curves) in the outer parts of galactic disks.

We are interested in the outer region of the galactic disk (in Fig. 1, this is the region with R > 10 kpc), where there is a large discrepancy between the Keplerian and observed rotation curves, and where the rarefied gas kinetics becomes decisive.

To build our model, we make two assumptions. Both assumptions play against us, i.e., we consider the worst option for us (this is important!).

- In developed model, I do not take into account active star formation at the edge of the optically visible part of galaxy. Since star formation in these regions leads to a depletion of the gaseous component (resulting in a density gradient directed outward of the galaxy), this simplification of the model is obviously working against us. Therefore, if star formation is correctly taken into account, the gas in the outer region of the galactic disk will be even more bounded in galaxy. In addition, it should be noted that we want to describe the RC at large distances from the center. However, star formation can noticeably affect only at the boundary of the outer part of the galactic disk, while its influence is negligible at large distances.
- I also will assume that in the outer part of the galactic disk, the gravitational interaction is negligible compared to the gas interaction. This is a reasonable assumption, as can be seen from Fig. 1. One can see that Keplerian curve drops dramatically with distance, while the observed rotation curve demonstrates the presence of some force much greater than gravitational force. I make this assumption (about the insignificance of the gravitational force) solely for the sake of simplicity of calculations, in order not to clutter up the calculations with unnecessary small corrections and insignificant details. Moreover, this simplification of our model also plays against us, because if gravitational force was considered, then the gas in the galactic disk would be even more bounded with the host galaxy. Thus, we again consider the worst case for us.

Considering all of the above, after simple transformations of expression (5), one can obtain the following equation, which gives the relationship between the tangential velocity (rotation curve), on the one hand, and the gas density, as a function of the distance from the center of the galaxy, on the other hand:

Having chosen such galaxies for which both measured column densities and rotation curves are available, one can (using this equation) implement one of the following two different ways: 1) take the measured column densities and using the resulting equation calculate the rotation curve, then compare them with observations, or 2) take measured rotation curves, substitute them into the resulting equation and calculate the column densities, which then should be compared with the observed ones.

In my work, I follow the second way. Two galaxies were taken, for which both rotation curves V(R) and measured column densities N(R) are available in literature. Based on the measured rotation curves, the column densities were calculated for this significantly different in size and masses galaxies (see Figs. 1.2.3.4 in the article). The coincidence with the observed distribution of gas density turned out to be excellent. This indicates the validity of the assumptions made and the correctness of the model. Thus, the “non -physical” rotation curves of galaxies are simply tailwinds of rarefied gas, the dynamics of which is completely described by gas kinetics. One can conclude that DM is not required to explain RC of S-galaxies.

A few words about young stars, born in the outer parts of galactic disk, should be said. These stars are formed from gas and, at the time of birth, receive its average momentum. Elementary calculations in the framework of classical mechanics show that such stars will be gravitationally bounded with host galaxy. However, they will follow not circular, but elliptical Keplerian orbits. When constructing a comprehensive model, it should also be considered that the speed of star formation is associated with the density of gas. If the density falls exponentially with the distance (and this is the case), then the star formation rate will also fall exponentially as distance increase. In other words, new stars in the outer part of the disk are almost not formed. It is likely that rare single stars observed in the outer parts of galactic disks were formed in the underlying layers and moving along elongated elliptical Keplerian orbits temporarily moved to where they are observed.

Summary.

Summarizing the discussed above, it can be argued that the rotation curves of the galaxies are formed without the presence of DM. To understand this, it is enough to assume that the rarefied gas obeys the laws of gas kinetics. In my modest opinion, the requirement of compliance with the obvious laws of physical kinetics is slightly less unusual than the requirement of the existence of DM in galaxies.

To model a spiral galaxy, three very different areas of galaxies can be separated:

1) R <R_0 (here R_0 is of the order of the value R_25; for a galaxy represented at Fig.1, it consists approximately 10 kpc.). This part of disk is populated mainly by stars (described as an ideal collisionless gas in gravitational field). Of course, the stars here are orbiting in consequence with the Kepler law. The influence of real gas is negligible due to its small abundance and the absence of interaction gas – stars.

2) R ≈ R_0. This is a transition area where an active star forming processes take place. For this reason, the gas is depleted, and for calculating the rotation curves, many subtle details (such as the structure of galactic arms, the star formation rate, the temperature of the gas, etc.) should be also included.

3) R > R_0. This area of our interest in which gravitational interaction no longer has a decisive value. Here, the rotation curves are determined by the gas kinetics and the rotation speed V_{┴} = V_K + V_d, where V_K is Kepler velocity (it is placed as a boundary condition at the distance R ≈ R_0), and V_D is the dynamic speed of the gas stream due to diffusion. In this outer part of galactic disk, the gravitational interaction between gas and galaxy is weak. In consequence with law of gas kinetics, the gas moves along a circular orbit, forming tailwind. It is this tailwind that forms the “non-physical” rotation curve.

In accordance with the above, the problem of DM can be considered as solved. The history of dark matter, which has more than 90 years, is over.