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Gas kinetics in galactic disk. Why we do not

need a dark matter to explain rotation curves of

spirals?

Lipovka A.A.

Department of Research for Physics, Sonora University,

83000, Hermosillo,Sonora, México

May 27, 2020

Abstract

In present paper, we analyze how the physical properties of gas aﬀect

the Rotation Curve (RC) of a spiral galaxy. It is shown, that the observed

non-Keplerian RCs measured for outer part of disks, and the observed

radial gas distribution are closely related by the diﬀusion equation, which

clearly indicates that no additional mass (Dark Matter) is need. It is

stressed, that while the inner part of the RC is subject of the Kepler

law, for the correct description of the outer part of the RC, the collisional

property of the gas should also be taken into account.

To conﬁrm this fact we suggest both quantitative estimations and exact

calculation to show how the outer part of the RC is related with the gas

density of the galactic disk. We argue that the hydrodynamic approach is

not applicable for the correct modeling of large-scale gas kinematics of the

galactic disk and more general diﬀusion equations should be used. From

our result it follows, that if the gas density is high enough (more than

$10^{-5}~cm^{-3}$, the RC for the outer part of the disk is formed by

“wind tails” of gas.

Proposed calculations are based on solving both the Kepler’s and the

Fick equations and was carried out for two edge-on galaxies NGC7331 and

NGC3198, for which the precise measurements of the gas column densities

and RCs are available. An excellent coincidence between the measured

column density of gas and that calculated from observed RCs is obtained.

On the basis of the obtained result, we calculate the total masses of the

NGC7331 and NGC3198. They consist $32.5 \times 10^{10}~M_sun$

and $7.3 \times 10^{10}~M_sun$ respectively. Consequences for cos-

mology are discussed.

Keywords: Dark Matter; Rotation Curves; Gravitational Potential; Mass of

spiral galaxy; gas kinetics; Mestel’s disk.

Pacs numbers: 47.45.Ab, 51.10.+y, 95.35.+d, 98.35.-a, 98.62.Dm, 98.62.Hr

1

1 Introduction

Diﬃculties in explaining the kinematics of celestial objects within the framework

of Kepler’s law were ﬁrst noted by James Jeans and Jacobus Kapteyn in 1922

and then conﬁrmed by Jaan Oort in 1932 and by Fritz Zwicky in 1933. To solve

the problem, it was assumed that there is some unobservable, invisible mass

aﬀecting the kinematics of the observed objects. It should be noted that these

observations were carried out in optics and therefore dark matter was the only

way out of this situation. Later, when radio astronomy methods were developed,

it became possible to observe the galactic disks in the radio wavelength range

as well. It turned out that the large-scale gas motions may also not obey the

Kepler’s law, and it was quite logical to assume that we are dealing with all

the same dark matter in this case. For more than 90 years, the DM nature has

been widely discussed by both astronomers and physicists, but no signiﬁcant

progress in this direction was achieved.

As is known, the need for the DM concept was mainly due to two prob-

lems: 1) Observed non-Keplerian Rotation Curves (RC)s of spiral galaxies and

2) The presence of additional invisible mass in clusters of galaxies, leading to

the observed gravitational bounding of clusters and also to anomalously large

gravitational lensing produced by the clusters. However, recently a signiﬁcant

progress has been achieved in ﬁnding of the missing mass in clusters, reported

by Kovács et al. (2018). Authors argue that the missing baryons reside in large-

scale ﬁlaments in the form of warm-hot intergalactic medium. It should also be

added here that Biernaux, Magain and Hauret (2017), when processing lensed

images, showed that neglecting the diﬀuse lensed signal leads to a signiﬁcant

overestimation of the half-light radius, and therefore to an overestimation of the

lensing mass value. For these reasons, it can be recognized that the second item

(excessive masses of the clusters) loses its urgency, whereas the RCs of spiral

galaxies remains the most intriguing manifestation of the DM.

But the situation with DM in the disks of S-type galaxies is even more

diﬃcult, since recently more accurate observational data was published, that

sheds light on the dark matter properties. Namely, it was shown that there is a

signiﬁcant correlation between the features of the galactic RC and corresponding

spiral structure of the baryonic component: "The dark and baryonic mass are

strongly coupled" (Mc Gaugh, Lelli & Schombert 2017; see also Sancisi 2004;

McGaugh 2004; Möller & Noordermeer 2006). It is diﬃcult (if at all possible) to

realize this coupling within the framework of the conventional DM paradigm in

which the DM is coupled with baryonic component by gravitation only, and the

distribution of the DM is described by spherically symmetric functions obtained

from numerical simulations discussed for example by Navarro, Frenk & White

(1996); Merritt, et al. (2006); Katz, et al. (2017); Di Cintio, et al. (2014a) and

Di Cintio et al. (2014b).

Moreover, on the one hand, as it was shown by Kroupa, Pawlowski and Mil-

grom (2012), cosmological models based on warm or cold DM are not able to

explain observed regularities in the properties of dwarf galaxies. On the other

hand, last year the rotation curves for highly redshifted galaxies were reported

2

and it was clearly shown that a large fraction of massive highly redshifted galax-

ies are actually strongly baryon-dominated (see Genzel et al. 2017, Lang et al.

2017, and references therein). These data contradict the generally accepted

scenario of galaxies formation on the DM halos.

It should be mentioned here also that reported discrepancies between values

of the Hubble constant observed at early and late cosmological time (Verde,

Treu, Riess 2019) clearly indicate a crisis of the ΛCDM model (Riess 2020).

This fact may require a revision of the ΛCDM model.

Search for DM in laboratories is also unsuccessful despite the unprecedented

eﬀorts of many international collaborations. The FERMI experiment designed

to search for annihilation of DM and anti-DM clearly shows negative result

announced by Albert, et al. (2017). As part of the XENON collaboration,

the radioactive decay of xenon-124 due to double-electron capture, which has a

half-life of 1.8×10

22

years, was detected (this indicates the highest sensitivity

of the method) but no signs of dark matter were found (Aprile E. et al. 2019).

SENSEI collaboration reports of world-leading constraints on dark matter —

electron scattering for masses between 500 keV and 5 MeV (see Abramoﬀ et al.

2019).

Summarizing, the unsatisfactory situation with the current explanation of

the rotation curves of spiral galaxies becomes obvious. All mentioned above

clearly indicate a serious problem with the naive simulation of the spiral galaxy

dynamics and the mass distribution, based only on the assumption of the domi-

nant role of gravitational interaction. Moreover, it suggests the need for revision

of actually used models and argues that more accurate modeling of the galactic

baryon component and adequate consideration of all reasonable physical eﬀects

are strongly required. At present, such simulations of the density distribution

are performed on the basis of the assumption of the overwhelming gravitational

force domination (see papers mentioned before). In this case, based on hydrody-

namic simulations, it is concluded that the eﬀect of gas kinetics on the rotation

curves formation is negligible. Thus, the properties of the gas were not taken

into account properly when the quasi-stationary structure of the gas disk was

modeled. By taking into account that the rotation curves for the most impor-

tant - the outer part of the disk are observed mainly in the molecular lines and

21 cm line of neutral hydrogen, it is became clear that the inﬂuence of collisions

of the hydrogen atoms and ions on the formation of the stationary gas ﬂuxes,

should at least be correctly assessed.

As far as we know, such an attempt (to include the gas properties into con-

sideration) was made by Mestel (1963). He proposed a toy model that includes

not only the gravitational interaction, but also some general physical properties

of the gas that forms the disk. Despite the roughness of the model of a homo-

geneous isothermal self-gravitating disk, Mestel managed to obtain a solution

characterized by ﬂat rotation curves. The toy model of Mestel was considered

by Jalali & Abolghasemi 2002, and recently by Schultz (2012), who showed that

to form ﬂat rotation curves in Mestel’s disk, much smaller masses are needed

than previously thought.

It is necessary to emphasize here the fact that Mestel disks with ﬂat rotation

3

curves are observed in the protoplanetary disks where rotation is not Keplerian

outside of the inner few AU in spite of the absence of the DM inside it (see Yen

et al. (2015a), Yen et al (2015b), Yen et al (2017)).

Unfortunately, the Mestel’s toy model suﬀers from some signiﬁcant draw-

backs (see Demleitner and Fuchs 2001)and probably for the reason it is not

widespread. In this case, a new, more detailed consideration that takes into

account all signiﬁcant physical properties of the gas should be developed.

At the ﬁrst glance, the hydrodynamic approach mentioned above, could

become the standard calculation method in this case. But it hardly can be

applied because of two reasons.

1) We are not interested in local small scale movements, on the contrary, we

need to calculate the global RC of external part of the quasi-stationary galactic

disk.

2) Hydrodynamic approach can not be applied because of the extremely low

density of gas resulting in the signiﬁcant free path lengths, that leads to a vi-

olation of the conditions of the hydrodynamic applicability L

k

>> l

fp

, which

must be superimposed to integrate the kinetics equations when we derive the

equations of hydrodynamics. Here L

k

is the characteristic size of the hydro-

dynamic calculations (corresponds to the size of step of the computation grid),

and l

fp

is the mean free path of the particle. For example the typical density

for outer parts of disk is 10

−3

−10

−4

cm

−3

leads to l

fp

= 3 −30pc.

For this reason, the hydrodynamic approach becomes inappropriate, since

the characteristic lengths L

k

on which the values (density, temperature, pressure

etc.) are changed considerably, become comparable or even less than l

fp

.

The only reliable and not cumbersome method can be obtained directly from

the kinetic equations the same way as it was made for hydrodynamic equations.

In our paper we are going this way.

Namely, the kinetic equations should be ﬁrst averaged over the volume in

order to avoid the limitations imposed on the applicability of hydrodynamic

equations, and only after such a procedure they can be integrated over the

momentum. Doing so we obtain diﬀusion equation with modiﬁed diﬀusion coef-

ﬁcient and evaluate the gas kinetic contribution to the RC formation. We prove

that to model a disk of a spiral galaxy, the gas kinetics should be taken into

account by correct way, because if the inﬂuence of the gas kinetics on the RC

formation is neglected, it will leads to a completely incorrect (pure Keplerian)

model of the galactic disk and, as a consequence, to a wrong estimation of the

gravitating masses and their distribution. We show that the solution of the

Fick equations strongly implies that the rotation curve of the gas does depend

on the gas density. Thus, if such a dependence is observed, this will be direct

evidence of the exceptional role of gas kinetics in the formation of the rotation

curves of galactic disks. With the example of two edge-on galaxies NGC7331

and NGC3198, we argue that the observed gas distributions do correspond to

the rotation curves and they are interrelated by the Fick equations as it should

be. Therefore, we show that it is the gas kinetics, that dominates in the forma-

tion of the rotation curves of spiral galaxies at large distances. We choose these

two edge - on galaxies to minimize the inﬂuence of the 3D structure of galaxy

4

on the column density, since we are interested in the disk component only.

Thus, we can conclude that the deviation of the observed rotation curves

of spiral galaxies from the Kepler’s law can be easily explained if the kinetics

of gas is correctly taken into account. In fact the RCs at large distances from

the center are just the wind tails of the baryon gas that follows the preceding

baryon matter in the case if the gas obeys the conventional laws of gas kinetics.

For this reason we do not need the DM concept to explain rotation curves of

spiral galaxies.

It should be noted here that a wind in galaxies is not a new concept. For

example a starburst-driven galactic wind in starburst galaxies is a well-known

phenomenon that is widely discussed in the literature, see for example Jones et

al. (2019), Sharp & Bland-Hawthorn (2010), Rekola et al. (2005) and references

therein. However, in the case of large-scale movements and extremely low gas

densities in outer part of disks that we are interested in, the standard hydrody-

namic approach cannot be applied because of the limitations mentioned above.

Thus, the wind which forms rotation curves, must be described in a diﬀerent

way.

The article is organized as follows:

In the second section a preliminary estimates are made for the main gas

parameters in the disk of a spiral galaxy. From the evaluations obtained here

one can see that the gas kinetics plays a dominant role in the formation of the

wind tails in the media out of equilibrium. But if so, the dependence of the outer

part of RC on the gas density should be clearly manifested. The most interesting

and convenient case for integration, corresponds to the constant RC for outer

part of the disk. For this reason in the third part we obtain diﬀerential equations

that describe the gas dynamics. By using the obtained diﬀusion equations we

calculate analytically the gas density as a function of distance from the center

of a galaxy, for the outer disk of the spiral galaxy in the case of constant RC.

While solving the diﬀusion equations gives us the volume abundance of the

gas, from observations we can only get the column density along the disk. For

this reason in the part 4 we calculate the column density to compare it with

the observed one. As an example we consider two edge-on S-type galaxies with

measured RCs and optical depth in 21 cm., as function of distance Rfrom

the center of the galaxy. We show that the results obtained with the Fick

equations, are in excellent agreement with the observational data. In conclusion

the main results of the paper are summarized and some important consequences

are discussed.

2 Preliminary estimates.

As it was mentioned above, there are two very diﬀerent components of galaxy

population: the stars and the gas, which are used to measure the RC of galaxy in

optics and in radio respectively. The ﬁrst component is driven only by gravita-

tion potential, whereas to describe the second one we should take into account

5

collisions and the gas kinetics should be involved in the consideration in the

right way.

To conﬁrm this fact, let’s evaluate some parameters of the gas. A rough

estimate of the mean free path time for a hydrogen atom t

fp

= (NσV

t

)

−1

(here

Nis the density of the gas in cm

−3

,σis cross-section for elastic collision

and V

t

is the mean thermal velocity of the atom) gives t

fp

≈1.3·10

10

/N

(sec) = 4.1·10

2

/N (yrs). For the typical HI density outside of the R

25

:

N≈10

−3

−10

−4

we obtain t

fp

≈4.1·(10

5

−10

6

)yrs.

However, as we know, the intergalactic gas, as well as the hot gas component

of the disk of a spiral galaxy is ionized. For this reason, we should estimate the

mean free path of the proton. The free path length (see Lang 1974) is:

l

∗

fp

=m

2

V

4

rms

z

2

1

z

2

2

N

e

e

4

ln Λ ≈3.2·10

6

T

2

z

2

1

z

2

2

N

e

ln Λ ,(1)

where z

1

e,z

2

eare charges of two interacting particles (for the proton and

electron we have z

1

=z

2

= 1) and N

e

is the electron density

Λ = 1.3·10

4

T

3

N

e

≈2·10

11

,(2)

so ln Λ = 26, and the proton’s mean free path is l

∗

fp

= 10

15

−10

16

cm for the

temperature T= 3000K. In this case the mean free path time can be evaluated

as t

∗

fp

=l

∗

fp

/V

t

= 10

9

−10

10

sec = 30 −300 yrs. As one can see these time (t

fp

or t

∗

fp

) are much smaller than the characteristic time of life of the galaxy, so the

collisions must be taken into account.

Thus, one can see that for description of the gas located in the outer R >

R

25

R

0

part of galactic disk, the complete gas dynamics equations should be

used to explain the observed RCs. Here R

0

denotes a distance to the transition

zone at which the contribution of the gas kinetics in formation of wind tails

(and hence RCs) begins to dominate, if compared with that caused by Kepler’s

law and R

25

is the radius at which the surface brightness of the spiral galaxy

falls to 25 mag arcsec

−2

in B-band.

Now let’s make another estimation to answer the question : Will the gas

be able to follow the underlying falling baryon matter to form the wind tails?.

From observations we know that baryon matter of a S-type galaxy moves along

a spiral (note that a galaxy is not a stationary object. it has it’s beginning, it

has the end, and it evolves over time, consuming intergalactic gas).

In this case we can imagine the underlying baryonic matter as a piston

(baryon matter in inner R < R

0

part of galaxy’s disk, the Kepler motion of

which is completely due to the inner baryonic mass). It moves in the spiral

tunnel with ideal walls (we can apply the homogeneous boundary conditions in

this particular case), and is followed by the HI gas (here we will not consider

the processes of star formation, that dilute the gas component, but we only note

here that accounting for such processes will increase the eﬀects under discus-

sion). The mean acceleration of the "piston" for typical galaxy can be roughly

evaluated as < w >= ∆V/∆t= (200 Km/s)/(10

9

yrs) = 10

−9

(cm/s

2

). By

6

taking into account the evaluation of t

fp

(or t

∗

fp

) made before, we can estimate

the variation of the piston’s velocity ∆Vduring the mean free path time of

the hydrogen atom. Even in the worst case of t

fp

for neutral component, we

have ∆V=< w > t

fp

= 10

−9

·2·10

10

/N = 20/N (cm/s). For typical density

N= 10

−3

(cm

−3

) we obtain ∆V= 2 ·10

4

(cm/s)<< V

t

≈10

6

(cm/s). So

one can conclude that even the neutral gas will follow the "piston" if the gas

density is high enough: N(cm

−3

)2·10

−5

·(10

6

/V

t

). We stress here that it

is important result which can qualitatively explain the great variety of the RCs

shapes because of their dependence on the gas density. From this estimate it

can be seen that when the density is small (N(cm

−3

)<2·10

−5

·(10

6

/V

t

)), the

wind tails will not be formed and the corresponding RC will decrease to coinside

with Keplerian one.

These were rather crude assessments, suggested here to show simplistically

the physics of the processes under diskussion. To conclude this part we would

like to stress that even in consequence with these simple estimations, one can

see that the gas, driven by collisions, will follow easily the underlying baryon

matter. Actually the gas under consideration forms the wind tail which is rigidly

follows the underlying baryon matter which, in turn, is driven mainly by the

gravity at distances R < R

0

. This way the absence of RC of S-type galaxies

in early universe, reported by Genzel et al. (2017), can be explained easily.

Namely, rough estimate of distance over which the wind tail (or, the same RC)

will spread is t·V

t

≈10

10

yrs ·3·10

7

·10

6

≈3·10

23

(cm) = 100 kpc. This

trivial evaluation clearly shows why the RC measured with HI line are seen in

our epoch, but can not be observed in early universe, characterized by the time

t < 10

10

yrs., as it was recently observed and reported by Genzel et al. (2017).

3 Gas density as a function of distance for con-

stant RC

At present, it is believed that collisions in the gas of a galactic disk can be

neglected even in the case of small galaxies (see, for example, Dalcanton &

Stilp 2010) and for large spirals it simply does not matter. Such conclusions

are based on hydrodynamic simulation and the assumption that the continuity

equation, the Euler equation, and hence the Bernoulli equation, are always

satisﬁed. However, this is not so in the case of the highly rareﬁed gas, and

therefore it is hardly possible to trust such calculations. Unfortunately, this

error is very common in hydrodynamic calculations applied to galactic disks,

the number of published works is huge and therefore we will mention here only

a couple of articles as an illustration of the problem under discussion: Joung

et al. (2009) ; Rosdahl et al. (2017). For example, the results of simulations

presented by Joung et al. (2009) clearly show that the characteristic scale

of the inhomogeneities, on which the gas parameters changes signiﬁcantly, is

L

k

= 1 −10 pc, while the mean free path of a particle at a density of 10

−3

(cm

−3

) is l

fp

= 3 −30 pc. As one can see, in this case the main condition for

7

the applicability of the hydrodynamic (HD) approach ( the Knudsen number

Kn << 1) is not fulﬁlled and HD can not be applyed. Let us consider the

calculation method in more detail.

It is well known that the hydrodynamic equations are obtained by integrating

kinetic equations over momentum. Kinetic equation for one type of particles is:

∂f

∂t +∂

∂x

α

(v

α

f) = Stf. (3)

Here fis the distribution function and Stf stands for the collision integral.

Integrating over momentum punder assumption that variations of all para-

meters ( density, velocity, temperature, etc.) are small at the mean free path

length l

fp

, i.e. the characteristic length L

k

>> l

fp

(or, which is the same - the

Knudsen number Kn =l

fp

/L

k

<< 1) , we can evaluate:

∂

∂x

α

(v

α

f)d

3

p≈∂

∂x

α

(v

α

f)d

3

p,

and obtain the equation of continuity:

∂n

∂t +∂

∂x

α

(V

α

n) = 0.(4)

Other hydrodynamic equations can be obtained by the same way. We will not

dwell on this now, referring the reader to the standard textbooks on physical

kinetics, but note here that all these equations also will suﬀer of the same

restrictions mentioned above:

L

k

>> l

fp

,(5)

which imposes some restrictions on the hydrodynamic equations applicabil-

ity (see Landau & Lifshitz v.X). Unfortunatelly many authors who investigate

numerically the kinematics of the galactic disks, do not bother to verify that

the conditions for the applicability of the equations of hydrodynamics (5) are

satisﬁed and violate these restrictions. Thus, the results obtained by them are

hardly credible.

To eliminate this restriction (5), we start with the same kinetic equation (3),

as it takes place in the case of hydrodynamics. To obtain eq.(4) we considered

condition (5) to be fulﬁlled. But if this is not the case, i.e. (v

α

f)is changed

signiﬁcantly within the scale l

fp

(this takes place in the case of an extremely

rareﬁed gas, when the distribution function is not well deﬁned and its derivative

strictly speaking does not exist), then the only we can do - is average over a

volume Ω(recall that we are not interested in small-scale gas motions and

therefore we can average) in order to redeﬁne the function (v

α

f).

Now we rewrite (3) as

Ω

∂f

∂t +∂

∂x

α

(v

α

f)d

3

xd

3

p= 0,(6)

8

and integrate it. First term gives the variation of the total number of parti-

cles N

tot

in the volume of integration Ω.

By using the Gauss theorem, the second term can be transformed to ∆S

α

∂N/∂x

α

(here ∆S

α

is the surface area of the integration volume, orthogonal to the parti-

cle ﬂow, and Nis the averaged density of the particles in the volume Ω, measured

in cm

−3

), so ﬁnally we have (see Appendix for details):

dN

tot

dt =−D∆S

α

∂N

∂x

α

.(7)

In this equation we introduce the modiﬁed diﬀusion coeﬃcient D=V∆l,

where Vis the bulk velocity of gas in Ω, and ∆lis the characteristic size of the

integration volume (∆l >> l

fp

). This is well-known diﬀusion equation.

Summarizing, we know that the hydrodynamic equations can not be applied

to simulate the dynamics of the rareﬁed gas. Moreover, we do not need this

approach to calculate RC in the case of an established quasi-stationary solution.

By integrating (3) over a volume and making the problem insensitive to the

small-scale inhomogeneities, we obtain the diﬀusion equation (7) suitable to

describe the large-scale movements of the rareﬁed gas in outer part of the disk.

In polar coordinates, expression (7) can be written as:

V

d

=−D

N

∂N

∂R , V

d⊥

=−D

N

1

R

∂N

∂ϕ .(8)

Now, we denote by R

0

the distance at which the Kepler motion ends and

the “unphysical” behavior of the baryonic matter (explained by introducing

sophysticatedly distributed dark matter) begins. Then the Keppler’s speed at

this distance is:

V

K0

=MG

R

0

.(9)

Consider the movement of a certain part of the gas. Let N(R, ϕ)be a

smooth, parametrizied function of the coordinate Rand ϕ. In this case, we can

write R=R(t)and ϕ=ϕ(t), where tis a parameter, dR =dR/dt ·dt and

dϕ =dϕ/dt ·dt. Taking into account that Rdϕ/dt =V

tot

⊥

, we obtain:

1

R

∂N

∂ϕ =−V

d

V

tot

⊥

∂N

∂R .(10)

For this reason from eqs. (8) it follows:

V

d⊥

V

tot

⊥

=V

2

d

.(11)

or

D

n

1

R

0

∂n

∂r =−V

d⊥

V

tot

⊥

),(12)

where we introduce n=N/N

0

,r=R/R

0

, and N

0

=N(R

0

).

9

Now calculate corresponding gas density distribution in order to compare

it with the observed distribution. By other words, we are interested in the

question: "which HI column density function corresponds to the case of the

constant rotation curve of baryon matter in absence of DM for an S-type galaxy

(in the case when the rotation curve is just the wind tail, the movement of which

is determined only by baryonic matter, without involving dark matter in con-

sideration)?" If the calculated density distribution coincides with the observed

one, it will be a serious argument against the presence of dark matter in the

disks of the spiral galaxies. Consider this problem in more detail. In order to

facilitate calculations, we consider a galaxy with a ﬂat rotation curvewe, i.e.

V

tot

⊥

= (V

d⊥

+V

K0

) = const.and √V

d⊥

V

K0

=const. By taking into account

the fact that D=const for very rareﬁed gas and integrating (12) we obtain:

n=n

0

exp −R

0

√V

d⊥

V

K0

Dr,(13)

where

n

0

= exp R

0

√V

d⊥

V

K0

D.(14)

As one can see this density distribution depends exponentially on the dis-

tance r. Unfortunately we are not able to realize a direct measurement of the

gas density of the galactic disk. The only we have are the observed column den-

sities measured in 21 cm., so now we are going to calculate the column density

formed by distribution (13).

4 Column density

By deﬁnition the column density is

N

L

= 2

L

0

Ndl, (15)

where Nis density function. By taking into account that l

2

=R

2

−ρ

2

, the

eq. (15) can be rewritten as:

N

L

= 2N

0

R

0

r

max

r=ρ/R

0

nrdr

r

2

−

ρ

2

R

2

0

.(16)

This integral can be estimated if we take into account that the density n

decreases exponentially with distance r.

Consider the distance ∆lat which the density drops by about 10 times. A

trivial estimate gives ∆R/R

0

≈2/κ, where

κ=R

0

√V

d⊥

V

K0

D.(17)

10

Straightforward calculationgives the following estimate for the integral (m.16):

N

L

(ρ

R

0

)≈N

0

R

0

κe

−κ(

ρ

R0

−1)

κρ

R

0

+ 2.(18)

The calculated column density (18) can be used to ﬁt the observed one

suggested by Begeman (1987, 1989) and Bigiel et al. (2010) for NGC7331 (ﬁg.1)

and NGC3198 (ﬁg.2).

Figure 1: Measured (squares) suggested by Begeman (1987); Bigiel et al. (2010)

and calculated with (24) (solid line) HI column density for NGC7331.

These two galaxies were chosen because they are seen edge on by observer.

Due to this circumstance, in this case there is no need to take into account the

angle of inclination of galaxy, that simpliﬁes the modeling.

As can be seen from the ﬁg.1 and ﬁg.2, the calculated (in assumption of ﬂat

RC) column densities perfectly ﬁt the observed ones for very diﬀerent galaxies,

characterized by diﬀerent mass and slopes of the column density function. So

we can conclude that RC at large distances R > R

0

are formed by wind tails

of gas which obeys the diﬀusion equation (7). the obtained relation connecting

the velocity and density through equation (7) clearly indicates that there is no

need to introduce dark matter into the model and we do not need dark matter

to explain the rotation curves of spiral galaxies.

11

Figure 2: Measured (squares) suggested by Begeman (1987); Bigiel et al. (2010)

and calculated with (24) (solid line) HI column density for NGC3198.

Now (by taking into account that RC consists of two parts: 1) R < R

0

where

the gravitation dominates, and 2) R > R

0

, where contribution of gas kinetics

becomes dominant) we can estimate the masses of two galaxies mentioned above

by using their measured RC suggested by Begeman (1987), Begeman (1989)

and de Blok et al. (2008), and previously obtained model for the baryon mass

distribution (see Lipovka 2018). The coeﬃcients α

∗

k

and β

∗

k

we immediately ﬁnd

from approximation of the pure baryonic RCs (R < R

0

) for these two galaxies

by using expansion suggested by Lipovka (2018):

V

2

⊥

=η

R

k

α

∗

k

β

∗

k

1−

3

2

β

∗

k

R

2

+ 1

(β

∗

k

R

2

+ 1)

3/2

,(19)

where the constant η= 2πG10

10

M

⊙

.

Figures 3 and 4 demonstrate results of such approximation for NGC7331

and NGC3198 respectively.

The thick straight horizontal line (the constant part of the RC) at the ﬁgures

3 and 4 corresponds to the wind tails (rotation curves) formed by gas which

obeys the diﬀusion equations and has column densities shown at ﬁgures 1 and

12

Figure 3: Measured (squares) as suggested by Begeman (1987, 1989), de Blok

et al. (2008), and calculated with the model of Lipovka (2018) (dashed line)

rotation curve for NGC7331. Wind tail (external part of RC) that corresponds

to the HI distribution (see ﬁg.1) is shown by the horizontal bold solid line.The

length of the line exactly matches the size of the ﬁg.1.

2. As it can be seen, the wind tails (constant RCs) extend exactly to the distance

where the column density function has the exponential form (18).

Obtained coeﬃcients for NGC7331 are α

∗

1

= 0.333 ,β

∗

1

= 0.077 ,α

∗

2

= 7.7

,β

∗

2

= 29.9, and for NGC3198 we ﬁnd α

∗

1

= 0.2,β

∗

1

= 0.26 ,α

∗

2

= 0.55 ,

β

∗

2

= 6.0. Now the masses of these galaxies can be obtained immediately with

relation for the total baryon mass suggested by Lipovka (2018). In this case for

NGC7331 we ﬁnd M

7331

= 32.5·10

10

M

⊙

and for NGC3198 the total mass is

M

3198

= 7.3·10

10

M

⊙

.

5 Conclusions and diskussion

In present paper we show that the commonly accepted explanation of the rota-

tion curves of spiral galaxies, based on the naive simulation of the spiral galaxy

dynamics and mass distribution in approach of the dominant role of the gravi-

tational interaction, is not complete and can not be considered as satisfactory.

We argue that the inﬂuence of the gas kinetics on the formation of rotation

curves is important, i.e. the physical properties of the gas must be taken into

account to determine the quasi - stationary structure of the gaseous disk named

13

Figure 4: Measured (squares) as suggested by Begeman (1987, 1989), de Blok

et al. (2008), and calculated with the model of Lipovka (2018) (dashed line)

rotation curve for NGC3198. Wind tail (external part of RC) that corresponds

to the HI distribution (see ﬁg.2) is shown by the horizontal bold solid line. The

length of the line exactly matches the size of the ﬁg.2.

as rotation curves (recall that the rotation curves for the most intrigue - the

outer part of the disk, are observed in the molecular lines and 21 cm. line of

neutral hydrogen). Therefore, the inﬂuence of collisions of the hydrogen atoms

and ions on the formation of the stationary gas ﬂuxes must at least be correctly

estimated. In our paper, we suggest such estimations. We show that the solution

of the Fick’s equations implies that the rotation curve of the gas and the gas

density are related by the Fick’s equations. Such dependence, if observed, will

proves the importance of the gas kinetics in formation of outer part of RC. We

consider as an example two edge-on galaxies NGC7331 and NGC3198. It turned

out that the observed gas distributions do correspond to the rotation curves and

they are related exactly by the Fick’s equations. Therefore, we conclude that it

is the kinetics of the gas that dominates in the formation of the rotation curves

of spiral galaxies at large distances and no DM needs to explain their extended

ﬂat RCs.

A couple of words should be said on the rare stars formed in the outer part

of the disk. It is known, the RC for outer part of disk is measured not only in

21 cm., or in molecular lines, but sometimes also in optics by using a spectra of

rare and young stars that were formed in this region. In this case approximately

the same (as the gas has) tangential velocities of remote rare stars that move

14

out of R

0

can be explained as momenta obtained from the gas of which the stars

were formed. Elementary estimates show that these young stars in most cases

will be bounded, but will move in elliptical orbits.

The main results of the paper can be summarized as follows:

1) It is argued that the hydrodynamic approach (like any other approach

based on the hydrodynamic description) is not applicable in the case of a rareﬁed

gas of the outer part of the galaxy disk. For this reason, to describe correctly

the observed RCs proﬁle of spiral galaxies, not only gravitation interaction, but

also the physical properties of the gas should be taken into account by using

correct model.

2) We show that RCs consist of two parts. One (inner part) is formed by

collisionless ideal "gas", consisting of stars, and in this case the gravitation

interaction dominates, whereas another part (localized in the outer region of

disk) is formed mainly by the real gas. In this case, the motion of the gas obeys

not only the gravity, but also the gas kinetics, which contribute to the formation

of the gas stationary ﬂuxes (and, consequently, to formation of outer part of the

observed RC). To model the gas movement, we derive the diﬀusion equation

(ﬁrst Fick law) with modiﬁed diﬀusion coeﬃcient.

3) On the basis of the Fick’s equations, the direct and exact relationship

between ﬂat RC and the density function n(R) for the gas is obtained. From

the measured rotation curves, we calculate the HI column density as function

of distance R for two edge-on spiral galaxies: NGC7331 and NGC3198. The

calculated column densities are in excellent agreement with the observed ones.

4) By taking into account the facts proved above, (that the RC consists of two

diﬀerent parts that are governed by Newtonian gravity and the laws of physical

kinetics of gases, respectively.) the total masses of two edge-on spiral galaxies

are calculated. Our evaluation for the NGC7331 is M

7331

= 32.5·10

10

M

⊙

and

for NGC3198 the total mass consists M

3198

= 7.3·10

10

M

⊙

.

In summary, it can be argued that there is no need for the introduction of

dark matter to explain the rotational curves of the S-type galaxies. The need

for DM arose from the use of an inadequate hydrodynamic model, which, due

to initial constraints, cannot be applied to calculate the dynamics of a rareﬁed

gas in the outer regions of galactic disks.

At ﬁrst glance, the absence of dark matter in nature can dramatically aﬀect

cosmological models because the DM is believed to play the key role in formation

of the observable structure in the universe. As is known, in the absence of dark

matter in the framework of the (pseudo) Riemannian Universe, the cosmological

time is not enough for the observed structures to be formed. In this case (in the

absence of DM) the only reasonable extension of the existing paradigm, which

satisﬁes the principle of the Occam’s razor, is the extension of the (pseudo-)

Riemannian geometry to the Finslerian one, that will give the necessary time for

the observed structure be formed. Actually the (pseudo-) Riemannian geometry

is a very special case of the Finslerian one, and there are no compelling reasons

for such a particular quadratic restriction.

Moreover, there are serious arguments in favor of the fact that we live on

the Finsler manifold, and not on the Riemannian one.

15

Firstly, only within the framework of the Finslerian geometry, the cosmo-

logical constant appears in a natural way from geometry itself, it has natural

explanation and it becomes possible to unify quantum theory and gravity (see

Lipovka 2014, Lipovka 2017).

Secondly, on the Finslerian manifold the Planck constant calculated from

the ﬁrst principles (with measured cosmological parameters) coincides with it’s

experimental value up to the second signiﬁcant digit, that is, to the measurement

errors of cosmological parameters (Lipovka 2017), whereas if it is calculated for

the (pseudo-) Riemannian world, we ﬁnd that the Planck constant diﬀers by

factor 3/2 from it’s exact value (Cardenas, Lipovka 2019). These are more than

serious arguments in favor of the Finsler geometry. If we also add to this the

observationally proven lack of dark matter in the early Universe and in clusters

of galaxies (see introduction), its absence in the disks of galaxies (this work),

then the need to move to the Finslerian world becomes obvious. In this case, of

course, the angles of gravitational lensing should also be recalculated using the

Finsler metric.

6 Acknowledgments

I would like to express my deepest gratitude to Dr. Gosachinsky I.V. for useful

discussions of observational data and the galactic gas physics.

The author devotes this work to the blessed memory of his teacher of diﬀer-

ential and integral calculus S.R. Tikhomirov.

7 Appendix A

We start form the eq. (6) of the paper:

Ω

∂f

∂t +∂

∂x

α

(v

α

f)d

3

xd

3

p= 0,(6)

and integrate it. This equation is similar to that we use to obtain the

continuity equation (4), but here the averaging over characteristic volume Ωwas

performed in order to avoid the restrictions Kn << 1that the HD approach

suﬀer of. For this reason eq. (7), obtained from (6) does describe movements of

gas, but does not suﬀer of the restrictions applied to the HD approach, discussed

in the manuscript.

Now let us consider transition from (6) to (7) in details.

First term gives the variation of the total number of particles N

tot

in the

volume of integration Ω:

Ω

∂f

∂t d

3

xd

3

p=∂N

tot

∂t ,(A1)

16

Consider second term of eq. (6):

Ω

∂

∂x

α

(v

α

f)d

3

xd

3

p, (A2)

By applying the Gauss theorem we obtain:

Ω

∂

∂x

α

(v

α

f)d

3

xd

3

p=

Σ

(v

α

f)d

3

pdσ

α

=

Σ

(V

α

N)dσ

α

,(A3)

Where Ωis the volume, Σis the corresponding surface, dσ

α

is a surface

element, V

α

is the bulk velocity in the volume, and Nis the density averaged

over Ω.

By taking into account that the velocity V

α

everywhere on surface dσ

α

, is

approximately constant and by applying the Gauss theorem, we can write:

Σ

(V

α

N)dσ

α

=V

α

Σ

Ndσ

α

=V

α

Ω

∂N

∂x

α

dσ

α

dx

α

,(A4)

Now, to obtain the linearized equations we suppose that Nis slowly changed

function of the coordinate and for this reason in the linear approximation we

can evaluate the integral as:

V

α

Ω

∂N

∂x

α

dσ

α

dx

α

=V

α

∆l∆S

α

∂N

∂x

α

=D∆S

α

∂N

∂x

α

,(A5)

Where ∆land ∆S

α

are the characteristic length and corresponding orthogo-

nal surface respectively and the coeﬃcient Dstands for V

α

∆l. Here ∆l >> l

fp

and therefore the restriction (5) is lifted.

Thus the ﬁnal equation can be written as

∂N

tot

∂t =−D∆S

α

∂N

∂x

α

,(7)

Here N

tot

is the total number of particles in the volume Ω, and Nis the

averaged density of the particles in the volume.

In the case when we have a diﬀusion velocity V

α

d

through the boundary

surface ∆S

α

, then the variation of the total number we can write as

∂N

tot

∂t =V

α

d

N∆S

α

,(A6)

and (7) becomes

V

α

d

=−D

N

∂N

∂x

α

,(8)

Thus we obtan equations (8).

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