_{1}

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This study contains the solution of the dark matter mystery of spiral galaxies by using the space of four distance dimensions
x
,
y
,
z
,
x
'
, in which x
'
is the fourth distance dimension. The calculation of galaxy rotation has been presented in the space of four dimensions by using two dimensional section x, x
'
and three dimensional section x, y, x
'
. The four dimensional mass M which generates the main gravitation field of galaxy is located at the fourth dimension at the distance x
'
= X
'
and other dimensions are zero x = 0, y = 0, z = 0. The method to calculate the speed distribution curve of four dimensional mass V_{M}: the speed distribution curve V_{M} is calculated by using the equation in which the gravitational force is equal to the centrifugal force
of rotation. The solution of this equation yields the speed distribution component V_{M} of the four dimensional mass M and the value of the mass M. In the publication
[1]
has been presented rotational speed distributions curves of the galaxy NGC 3198. The speed distribution curve of galactic halo in that publication corresponds to the speed distribution curve V_{M} of four dimensional mass M of this study. In order to find out how well this four dimensional model functions, the speed distribution curve V_{M} of four dimensional mass M has been calculated by using two pairs of rotational radius and speed values. The conclusions and findings: the calculated distribution curve V_{M} was a good match for the halo curve of the publication
[1]
. Furthermore, four rotational speed distribution curve V_{M} was calculated using different values of the distance X
'
, which yielded different values for the maximum radius of galaxy. In this manner the different galaxy models of the publication
[2]
^{}
^{ }were obtained. By that means the solution of dark matter mystery has been proved.

The Sun and other stars which are visible in the sky are a part of the Milky Way galaxy. The Milky Way is a typical spiral galaxy which consists of a thin rotating disk of stars, and in its middle is a bulge of dense located stars. The stars of the Milky Way rotate round about the center of galaxy at a certain rotational speed. The Sun and the earth are located in the middle of the disk of the Milky Way at the distance of 25,000 light years from its center, and they rotate round the center at about the speed of 220 km/s. Edwin Hubble was the first astronomer who inferred that spiral nebulae are galaxies which are located at great distances from the Milky Way, publication 1929. In publication [

In the following, a simple method has been presented which makes possible to see the space of four distance dimensions, and to do calculations in it. In

In

It seems that this model fits very well for the measurements. Near the center of galaxy, rotational speeds from the redshift measurements and theoretically calculated speeds from the light intensities of rotating stars have relatively good correspondence. In the center of galaxy, the gravity force between the star mass m and the four dimensional mass concentration M is in the x' axis direction, and it disappears completely. At the border areas of the galactic rotating disk the gravity force component in the x axis direction Fcosα increases which is according to the redshift measurements. Even over 90 % of the gravity forces acting at the border areas of galaxy cannot be explained by the masses of the visible stars. This model of the space of four dimensions can explain the weird rotation speeds of galaxies.

The mystery of dark matter in the rotation of galaxies is a common subject matter in the study books of cosmology, publication [

In the following star’s rotational speed v round the center of galaxy has been derived. The speed components of star’s rotational speed v: speed component from visible matter v_{n}, ordinary dark matter v_{p}, four dimensional mass M v_{M}. The differentia of the gravitational force is equal to the differentia of the centrifugal force.

γ m Δ M ( x ) ( R − x ) 2 = m R d ( v n 2 ) (1)

Summation yields the following equations.

γ ∑ m Δ M ( x ) ( R − x ) 2 = m ∫ d ( v n 2 ) = m v n 2 R (2)

The centrifugal force components are from visible mass, ordinary dark mass and four dimensional mass M.

m v n 2 R , m v p 2 R , m v M 2 R (3)

The following equation is obtained by adding the three centrifugal force component.

m v 2 R = m v n 2 R + m v p 2 R + m v M 2 R (4)

or

v 2 = v n 2 + v p 2 + v M 2 (5)

Total rotational speed curve v and rotational speed curve of visible mass v_{n} are known, rotational speed curve of ordinary dark matter v_{p} is not knows, but it can be concluded that it is the same form as the speed curve of visible matter. The rotational speed curve v_{M} of the four dimensional mass M is not known and it will be calculated by using the knowledge of the rotational system. In

F = γ m M ( R 2 + ( X ′ ) 2 ) 3 / 2 (6)

In which the component in the direction of x-axis is

F cos α = R γ m M ( R 2 + ( X ′ ) 2 ) 2 (7)

The centrifugal force component corresponding to the four dimensional mass M is

m v M 2 R (8)

Calculation of the speed distribution curve of the four dimensional mass M. The gravitational force is equal to the centrifugal force.

m v M 2 R = R γ m M ( R 2 + ( X ′ ) 2 ) 2 (9)

The speed component of the four dimensional mass M effecting on the star and the mass M

v M = R 2 γ M ( R 2 + ( X ′ ) 2 ) 2 (10)

M = v M 2 ( R 2 + ( X ′ ) 2 ) 2 R 2 γ (11)

Equation (9) at distances R_{1} and R_{2} from the center of galaxy

m 1 v M 1 2 R 1 = R 1 γ m 1 M ( R 1 2 + ( X ′ ) 2 ) 2 (12)

m 2 v M 2 2 R 2 = R 2 γ m 2 M ( R 2 2 + ( X ′ ) 2 ) 2 (13)

By dividing the two equations above

R 2 2 v M 1 2 R 1 2 v M 2 2 = ( R 2 2 + ( X ′ ) 2 ) 2 ( R 1 2 + ( X ′ ) 2 ) 2 (14)

a = v M 1 R 2 v M 2 R 1 (15)

a ( R 1 2 + ( X ′ ) 2 ) = R 2 2 + ( X ′ ) 2 (16)

Distance of the four dimensional matter M from the center of galaxy

X ′ = R 2 2 − a R 1 2 a − 1 (17)

Values of total rotational speed distributions v have been calculated from redshifts of rotating stars of galaxies. Rotational speed distribution component of visible stars v_{n} has been calculated from the gravity force of their mass, which has been calculated according to their light intensity, the result of which has been like the distribution curves in _{p} is the same form as the speed distribution curve from visible light v_{n}. By using these three speed distribution curves it is possible to calculate with Equation (5) the speed distribution curve which corresponds to the speed distribution curve v_{M} of four dimensional matter M. In the publication [_{M} of Equation (10). They can be compared. The system of galaxy rotation of this study is a hypothesis, and it must be proved to be true or not true. With two pairs of values of rotational speeds v_{M}_{1} and v_{M}_{2} and radii R_{1} and R_{2} it can be done.

In the publication [

publication corresponds to the speed distribution curve v_{M} of four dimensional mass of this study. In order to find out how well this four dimensional model functions, the speed distribution curve v_{M} of four dimensional mass has been calculated by using these rotational radius and speed values.

R 1 = 3 × 10 20 m , R 2 = 8 × 10 20 m

v M 1 = 80 km / s , v M 2 = 130 km / s

Distance of the four dimensional mass M from the center of galaxy, Equation (15) and Equation (17)

a = v M 1 R 2 v M 2 R 1 = 0.8 × 8 1.3 × 3 = 1.64

X ′ = R 2 2 − a R 1 2 a − 1 = 8 2 − 1.64 × 3 2 0.64 × 10 20 m = 8.8 × 10 20 m

Four dimensional mass M, Equation (11)

M = v M 1 2 ( R 1 2 + ( X ′ ) 2 ) 2 R 1 2 γ = 0.8 2 × 10 10 ( 3 2 × 10 40 + 8.8 2 × 10 40 ) 2 3 2 × 10 40 × 6.67 × 10 − 11 kg = 79.7 × 10 61 kg

From Equation (10) the effect of four dimensional mass M on the rotational speed distribution curve v_{M} is obtained as follows:

v M = R 2 γ M ( R 2 + ( X ′ ) 2 ) 2 = M ⋅ R 2 γ ( R 2 + ( X ′ ) 2 ) 2 (18)

As it is obvious that the effects of three and fourth dimensional masses are the same, except for the effect of the distance R, it can be inferred that the coefficients of gravity are the same, except for the quality, γ = 6.67 × 10^{−11} N∙m^{3}∙kg^{−2}.

The total rotational speed distribution of galaxy NGC 3198 has been presented in the publication [_{n} + v_{p} in this study, and rotational speed component of galactic halo which corresponds in this study the rotational speed component v_{M} of four dimensional matter M, Equation (10).

1) In order to find out how well this four dimensional model functions, the speed distribution curve v_{M} of four dimensional mass M has been calculated by using two pairs of rotational radius and speed values which are approximately the same as the speed curve values of galactic halo of the publication [_{M} of four dimensional mass M can be compared to the corresponding speed curve of real galaxy rotation. By using these values it was obtained the four dimensional mass value M = 79.7 × 10^{61} kg, Equation (11) and the distance above the center of galaxy in the four dimensional axis X' = 8.8 × 10^{20} m, Equation (17). The rotational speed distribution component of the four dimensional matter v_{M} was calculated from Equation (10), and it is presented in

The other source of inaccuracy is the redshift measurements and the evaluation of the mass distribution of ordinary dark matter. The halo speed distribution curve of the publication [_{M} until the radius R = 30 kpc. In the publication [_{M} begins to be separated from each other significantly. The speed curve of four dimensional matter v_{M} decreases in this region, but the halo speed curve of the publication [_{M} begins to decrease.

2) The generation process of galactic system may be explained like this: The force of gravitation of the four dimensional mass M acts on the rotational plane of galaxy. It has no component at the direction of the fourth dimension because the three dimensional mass of galaxy of stars, planets, gas, dust and other matter have not gravity force of fourth dimension. The result is that the four dimensional mass M generates a gravitational field which has a great hole at the center of galaxy. The gravitational field of the four dimensional mass accelerates three dimensional mass of stars, planets, dust, and other matter into the speed of rotation in which it rotates round about the center of galaxy. In this manner the hole in the gravitational field of the four dimensional mass M fills up, and the typical constant speed outer boundary regions of galaxies have been generated. The gravity field of the four dimensional matter M accelerates a star to the rotational speed somewhat above 130 km/s, in which case it retain rotating the galaxy in the border region, and if the star loses kinetic energy and speed, it begins to rotate the galaxy at the center region, and if the star accelerates considerably more than 130 km/s, then the gravity force of the four dimensional matter cannot hold it, and it moves out of the gravity field.

3) In

4) Justification: Fitting the four dimensional galaxy rotation system to the measured values of the galaxy rotation.

Point 1. The speed distribution curve of four dimensional mass v_{M} is a good fitting to the corresponding measured halo speed distribution curve of publication [

Point 2. The generation of the galaxy system can be inferred.

Point 3. The calculation of the speed distribution curve of four dimensional

mass v_{M} with different mass M and distance X' values generate the measured speed distribution curves of galaxies in publication [

The author declares no conflicts of interest regarding the publication of this paper.

Rahikainen, A. (2020) Galaxy Rotation in the Space of Four Distance Dimensions. World Journal of Mechanics, 10, 83-94. https://doi.org/10.4236/wjm.2020.107007

Dimensions of ordinary space, x, y, z

Fourth distance dimension, x'

Four dimensional mass, M

Ordinary visible galactic mass, M(x)

Location of the mass M on the fourth distance dimension, X'

Rotational speed distribution curve, v

Rotational speed distribution component of the mass M, v_{M}_{ }

Rotational speed distribution component of visible mass, v_{n}

Rotational speed distribution component of ordinary dark matter, v_{p}

Radius of rotation of the galactic mass, R

Mass of a star, m