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Gas kinetics in galactic disk. Why we do not need a dark matter to explain rotation curves of spirals?

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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 x 10^{10} M_sun and 7.3 x 10^{10} M_sun respectively. Consequences for cos-mology are discussed.
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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
:
N10
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 (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
 
∂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 .
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 , measured
in cm
3
), so finally we have (see Appendix for details):
dN
tot
dt =DS
α
∂N
∂x
α
.(7)
In this equation we introduce the modified diffusion coefficient D=Vl,
where Vis the bulk velocity of gas in , 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/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:
 
∂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 was
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 :
 
∂f
∂t d
3
xd
3
p=∂N
tot
∂t ,(A1)
16
Consider second term of eq. (6):
 
∂x
α
(v
α
f)d
3
xd
3
p, (A2)
By applying the Gauss theorem we obtain:
 
∂x
α
(v
α
f)d
3
xd
3
p= 
Σ
(v
α
f)d
3
pdσ
α
=
Σ
(V
α
N)
α
,(A3)
Where is the volume, Σis the corresponding surface,
α
is a surface
element, V
α
is the bulk velocity in the volume, and Nis the density averaged
over .
By taking into account that the velocity V
α
everywhere on surface
α
, is
approximately constant and by applying the Gauss theorem, we can write:
Σ
(V
α
N)
α
=V
α
Σ
N
α
=V
α
∂N
∂x
α
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
∂N
∂x
α
α
dx
α
=V
α
lS
α
∂N
∂x
α
=DS
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 =DS
α
∂N
∂x
α
,(7)
Here N
tot
is the total number of particles in the volume , 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
NS
α
,(A6)
and (7) becomes
V
α
d
=D
N
∂N
∂x
α
,(8)
Thus we obtan equations (8).
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