Galaxies appear to rotate more quickly than allowed by the gravity of its observable matter. Logically, they should have been destroyed a long time ago, as should galaxy clusters. Because of this disagreement between theory and observations, some undetectable element is believed to be providing galaxies with the additional mass that they need to avoid breaking up. This mysterious element has been called “dark matter”. Over the years, coherent theories on dark matter and dark energy have been formulated to explain the stability of galactic systems. Attempts have been made to modify Newton’s laws of gravity, for example, Milgrom’s MOND theory [1] , which explains rather well the flat rotation curves of stars for many galaxies. However, these theories have not yet offered a sufficiently well explained source; they are rather empirical theories. In this article, using Newton’s laws and Einstein’s theory of general relativity [2] , I show that a coherent theory is possible that can explain, at the same time, flat rotation curves, the expansion of the universe, and the acceleration of the expansion. I will define a new term, , one limit of which is the Hubble constant, and I clearly show its relationship with the flat rotation curves of galaxies. This new term is universal because it applies without restriction to all celestial bodies.

The Hubble constant is the constant of proportionality between the distance and speed of apparent recession of galaxies relative to any point in the observable universe. It is connected to the famous Hubble’s law, which describes the expansion of the universe. Although referred to as a constant, actually varies depending on time and distance. It thus describes the growth rate of the universe at one moment and at a given distance from the point of observation. The most precise value of (from optical observations, May 2001) is 72 ± 8 km∙s⁻¹∙Mpc⁻¹.

However, the current definition of does not relate expansion of the universe to the flat rotation curves of galaxies and galaxy clusters. I must thus extend the concept of expansion of the universe to a comparison of the gravitational acceleration for small and large distances. Thus Newtonian acceleration can be written as for small distances (less than kpc), and and (for distances greater than kpc) are the measured acceleration and distance, respectively, between gravitationally interacting masses. We can therefore write the following relationship for a galactic system in gravitational balance:

where is the measured speed of rotation. Einstein’s relativity theory may also be applied to Newtonian gravitational acceleration so that the masses vary as

and the distances in the moving reference frame (for the observer) shorten according to. In fact, Newton’s second law in relativistic mechanics (special

relativity) may be applicable in my calculations. Here, represents the radial velocity between two gravitationally interacting objects (not to be confused with the rotation speed of planets, stars, galaxies, and galaxy clusters around the center of their respective system). This radial velocity can also be regarded as the recessional velocity of the object of interest (galaxies, stars, planets) similar to the relationship, where is the Hubble constant and D is the distance.

On the other hand, I take into account the relativistic function of acceleration developed by Elbeze [3] [4] , in which I include the recessional rate of the universe defined by function (added at the possible radial speed), which will be defined below (and is equivalent to the Hubble expression), and obtain:

From these relationships, I can define a rate of variation of acceleration between the Newtonian acceleration (for all distances) and the measured acceleration as follows:

I can compare Equation (3) with the Hubble constant as follows:

where is the equivalent of in this article, and is the reference radial velocity, which I will define below. By replacing accelerations and with their equivalent in speed and distance in Equations (1), (2), (3), and (4), so that radial velocity is negligible in calculating the rate, while applying the relativistic principle for distance and masses in radial motion (defined above and according to Newton’s second law in special relativity to a radial dimension along), and with

, I obtain:

where is the recessional speed of mass M₂ from mass M₁ (Figure 1).

Here, D is the distance between masses M₁ and M₂, their relative speed (radial velocity) is negligible, and is the measured speed of rotation of M₂ around mass M₁. Solving Equation (4) and Equation (5) for gives two universal solutions:

We can already assume that M₁ and the gravitational constant G (from equation 2) disappear from Equation (6), because they are factors in the elements com-

posing it. Only speeds and, distance D, and the reference speed V₀, which will be defined low, are present in Equation (6). Thus Equation (6) is universal, independent of M₁ and M₂, and applies to all points in the universe with no distinctions. Let us write the recessional velocities and as follows:

One possible solution for the speed of rotation, which solves Equation (6), is:

where is the acceleration of the expansion of the universe (which is here regarded as small compared with gravitational acceleration for distances < 1 Mpc); G is New- ton’s gravitational constant; M_g is the mass of the planet, star, galaxy, or galaxy cluster; D and are the distance and relative radial speed, respectively, between masses M₁ and M₂ (Figure 1); is the recessional velocity of the object of interest; is the atomic radius of hydrogen; and the two functions (value between 1 and 2) and, which is on the order of depending on the nature of the celestial body (e.g., galaxy, star), are defined in Appendix 1 and Appendix 2, respectively. By replacing

(Equation (8)) in Equation (6), the term from Equation (8) (which defines the Newtonian part of) disappears before the factors, so that with

then Equation (5) becomes

I can apply Equation (8) (e.g., to our galaxy) in the case of gravitational balance of the Milky Way, and assume the acceleration of the expansion of the universe to be equal to (defined low), here is the acceleration in, we obtain the Figures 2-4.

In Figure 2, the position of our Sun is at approximately 8.2 kpc and its speed is. The speed returns to its Newtonian form to 2.5 Mpc. In my calculations, only the mass of the bulb is considered; I do not take account of the mass of the galactic gas clouds, which in any case do not influence the effective existence of the flat rotation curves of the galaxy, but modify only the form of the curve speeds. I apply Equation (9) to define (the proposed new expansion rate) in relation to the Hubble constant (Hubble expansion rate) (Figure 3).

Figure 3 shows four terms:, the proposed expansion rate, and the Hubble constant. Values of are given in units of km・s⁻¹/Mpc; in this article, is equal to km・s⁻¹/Mpc for a distance of [5]

[6] , and for a large distance () is equal to km・s⁻¹/Mpc. I stated the position of the Sun in the Milky Way, which is approximately equal to 8.6 pc (0.0000068 Gpc) from the galactic center, with a rotation speed of 230 km∙s⁻¹ and an expansion rate of.

In Figure 3, we can see the difference between and. For, the speed represents a straight line with a fixed slope, whereas for, the slope is variable all along distance D. At large distances, the expansion rate is positive (repulsion, i.e., galaxies move apart) and tends toward the speed of light c (thus conforming to Einstein’s relativity theory); at a smaller distance it becomes negative (attraction) and almost constant, for part of the curve after the reversal since the distance. This part of the curve is at small distances less than, particularly regarding the solar system and its planets for which the expansion function also applies. I can also plot the curve of the acceleration of the expansion of the universe, taking the derivative of with respect to time as

where (Figure 4).

In Figure 4, we can see three areas of acceleration: a zone from 0 to 0.5 pc where there is a constant rate of negative expansion (of, attraction); a zone between 0.5 pc and 2 Gpc where there is an accelerated rate of positive expansion (repulsion); and a zone between 2 Gpc and above 3 Tpc where there is a decelerated rate of expansion (always repulsive). Finally, the acceleration produced by the expansion of the universe is much less than the acceleration produced by mass up to the value of 2.9 Mpc for the Milky Way, and is negligible for our solar system. It seems that beyond 2.9 Mpc, other celestial bodies cannot be orbiting the Milky Way!

I can rewrite an approximation of Equation (8) by neglecting the acceleration of the universe for the Earth (Figure 4) and by setting as follows (here for our solar system):

This reference speed V₀ will be calculated at the level of our solar system. This has two principal advantages: we know the precise Earth-Sun distance, and the flat expansion velocities are relatively stable (here attractive to the Sun; see Figure 3). Using universal Equations (8), (9), and (10), which also apply to our solar system, I deduce the value of, depending on the position and speed of rotation of the Earth around the Sun (and, respectively). By comparing the value of, which is given by Equation (10), with the value in Equation (9), I get:

In this relationship the value of is obtained by replacing the value of V₀ in Equation (9) by. Indeed, the V₀ value is similar at the speed of expansion of the universe, and this fact answers the classical formula of Hubble. The function is a constant dependent on the zone of space considered. For a distance D (about the distances of the solar system). Finally, the calculation of V₀ in the solar system gives. To obtain this value, I have to check the equation

where (Figure 5).

The term depends on the nature of the star, galaxy, or planet in question; here for our solar system, and for the Milky Way . The values of n and no define the function. Thus, the reference speed V₀ will have a value dependent on the nature of the considered system, or:

For the solar system:.

In the Milky Way galaxy:.

Thus, is a universal speed of reference for all gravitational systems. This value is thus directly related to the value of the expansion rate of the universe, as well as to the flat rotation curves of galaxies. Figure 6 shows the point where the distance of the Sun from the center of the Milky Way and the ratio

coincide. This equality requires the conditions and to the center of the Milky Way equal to.

I have shown in this article that it is possible to link the concept of expansion of the universe through a new term (which is comparable to the Hubble constant), and the flat rotation curves of galaxies, with one limit according to galactic mass. For the Milky Way, the limit (end of the non-Newtonian rotation curves) is around 2.9 Mpc. This is possible using both the widely accepted gravitational theories of Newton and Einstein. I suggest a solution (Equations (8) and (10)) for the expression of the flat rotation speed of galaxies (or other galactic objects) which is derived from the universal relation of expansion of the universe (Equation (9)). This solution gives excellent results. The first and second derivatives of the value of (Equation (6)) are a function of time, and show that the universe is in acceleration until about, then in deceleration until approximately. This is new information about the universe. Thus, we can legitimately ask whether dark matter is necessary to explain the flat rotation curves of galaxies? And because of this accelerating expansion, could it be that dark energy does not exist? In this article, I have shown that the basic concept of the expansion of the universe also applies to the solar system like with planets and their satellites. Based on this, the Hubble constant could be measured with more accuracy. The results presented in this article may solve some current issues such as the energy density of galaxies, their masses, and may help to solve the mystery of the formation of the universe and the presence of filaments of material connecting the clusters and super clusters of galaxies of the current standard model, all without the need to involve hypothetical dark matter and dark energy.

Elbeze, A.C. (2017) Hubble’s Constant and Flat Rota- tion Curves of Stars: Are Dark Matter and Energy Needed? Journal of Modern Physics, 8, 24-34. http://dx.doi.org/10.4236/jmp.2017.81003

Let, a relative coefficient taken as a reference for our galaxy, the Milky Way. Any other object in the universe will also have a number that varies according to the object’s characteristics such as its mass distribution, its linear density, and the shape of its flat rotation curves. For the Sun. In this article, the distribution of the masses is comparable to that of a thin disk (the spherical distribution does not change the final results) of apparent mass equal to:

where D is the distance from the center of the object of interest (e.g., galaxy, star) and is the radius of the central core, where for a galactic disk such as that of the Milky Way, is generally taken as equal to, for the sun and the earth their radius. The density is equal to.

The value of is defined as, where is the hidden value of the galactic mass near to its actual value for the distance being considered. Let this mass be equal to

I define the coefficients as being equal to

and finally obtain as a function of distance.

In this article, the value of for our purposes is

Here, is a function of at least three parameters: is the atomic radius of hydrogen; is the average atomic radius in the galaxy (or other object of interest) in terms of the orbital radius of hydrogen, where is defined in Appendix 1; and is the electron rest energy.

The relationship giving is written as follows:

The value is always very small compared with and (proton and electron rest energy), and we can write:

where and are the electron and proton rest energies, respectively; and are modified wavelengths of the electron and the proton; and, are the radial and volumetric interaction energy between the electron and the proton in the hydrogen atom. These values define the volume and the binding energy of the electromagnetic reaction between the electron and the proton, and and are the energy charge (similar to electric charge) of the proton and electron in the hydrogen atom as follow table:

By using the values in table, we obtain a value for our galaxy the Milky Way of. For a more detailed explanation of these values, the reader may wish to refer to the book by Elbeze [7] .

In fact, for a first reading of this appendix, we can neglect the values of the above table and consider only the value of as being a constant of integration.

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